A quintic B-spline technique for a system of Lane-Emden equations arising in theoretical physical applications

Osama Ala'yed; Ahmad Qazza; Rania Saadeh; Osama Alkhazaleh; Osama Ala'yed; Ahmad Qazza; Rania Saadeh; Osama Alkhazaleh

doi:10.3934/math.2024225

AIMS Mathematics

2024, Volume 9, Issue 2: 4665-4683. doi: 10.3934/math.2024225

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Research article

A quintic B-spline technique for a system of Lane-Emden equations arising in theoretical physical applications

1.
Department of Mathematics, Jadara University, Irbid 21110, Jordan
2.
Department of Mathematics, Faculty of Science, Zarqa University, Zarqa 13110, Jordan
3.
College of Humanities and Sciences, Thomas Jefferson University, East Falls Campus, Philadelphia, PA, USA

Received: 19 November 2023 Revised: 28 December 2023 Accepted: 29 December 2023 Published: 19 January 2024
MSC : 34K34, 34K32

In the present study, we introduce a collocation approach utilizing quintic B-spline functions as bases for solving systems of Lane Emden equations which have various applications in theoretical physics and astrophysics. The method derives a solution for the provided system by converting it into a set of algebraic equations with unknown coefficients, which can be easily solved to determine these coefficients. Examining the convergence theory of the proposed method reveals that it yields a fourth-order convergent approximation. It is confirmed that the outcomes are consistent with the theoretical investigation. Tables and graphs illustrate the proficiency and consistency of the proposed method. Findings validate that the newly employed method is more accurate and effective than other approaches found in the literature. All calculations have been performed using Mathematica software.

Keywords:

Citation: Osama Ala'yed, Ahmad Qazza, Rania Saadeh, Osama Alkhazaleh. A quintic B-spline technique for a system of Lane-Emden equations arising in theoretical physical applications[J]. AIMS Mathematics, 2024, 9(2): 4665-4683. doi: 10.3934/math.2024225

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Abstract

1. Introduction

Any Lie group is equipped with a natural linear connection $\nabla$ , and therefore, a canonical system of geodesic equations. This connection was introduced in 1926 by Cartan and Schouten ^[1]. Recently, a lot of work has been done on the symmetries of the geodesic equations of the canonical connection. Ghanam and Thompson considered the problem for all three and four-dimensional indecomposable Lie algebras ^[2]. They also considered six-dimensional nilpotent Lie algebras ^[3]. Almusawa et al. ^[4] considered the probelm for the five-dimensional indecomposable Lie algebras with co-dimension one abelian nilradical.

Recently, Almutiben et al. considered the problem for the case where the nilradical is of co-dimension two. In dimension four, there is only one such indecompsable Lie algebra with co-dimensional two nilradical, namely, $A_{4, 12}$ in the Winternitz list ^[5]. In dimension five, there are three five-dimensional Lie algebras with co-dimension two abelian nilradical. These algebras are $A_{5, 33}$ – $A_{5, 35}$ in ^[5]. In all these cases, a comprehensive analysis of the symmetries of the geodesic equations was performed. Almutiben et al. ^[6] also considered the problem for the six-dimensional solvable indecomposable Lie algebras. Following the classification given by Turkowski ^[7], there are forty classes of non-isomorphic six-dimensional Lie algebras. Among these forty algebras, the first nineteen $A_{6, 1}$ – $A_{6, 19}$ have a four-dimensional, or equivalently co-dimension two, abelian nilradical and a two-dimensional abelian complement. Almutiben et al has given a comprehensive analysis of the symmetries in these nineteen cases ^[6].

In this paper, we continue to study the symmetries corresponding to the eight algebras $A_{6, 20}$ – $A_{6, 27}$ in ^[7]. These algebras are characterized by the property that they have a four-dimensional abelian nilradical and a one-dimensional center.

An outline of the paper is as follows: In Section 2, we provide some background material that helps to motivate our analysis. We do not give very specific details, but do provide some useful references. In Section 3, we give the definition of the canonical connection $\nabla$ on a Lie group and a summary of its properties. In Section 4, we review the symmetries of differential equations and the Lie invariance condition. In Section 5, for each algebra $A_{6, 20}$ – $A_{6, 27}$ in Turkowski's list, we give the geodesic equations, a basis for the symmetry algebra in terms of vector fields, and, finally, we identify the symmetry Lie algebra in terms of the nilradical and its complement. We will use $\rtimes$ to denote a semi-direct product and $\oplus$ for the direct sum of algebras.

2. Background concepts

In order to motivate some of the material, we shall sketch a few of the key ideas encountered below. We shall be considering certain systems of second order ordinary differential equations. The space of independent variables that occur serve as a system of local coordinates on a Lie group $G$ . We shall take for granted the basic definitions and properties of Lie groups. One may think of a Lie group as being an object that is intermediate between a vector space and a differentiable manifold. In particular, on a Lie group, one may make sense of various geometric objects (vector fields and one-forms primarily) as being left or right invariant. We refer the reader to ^[8,9,10] for readable introductions to the topic, that will be helpful in understanding the present article. In addition, these references help to explain the relationship between Lie groups and Lie algebras in a pragmatic way. Although the differential equations treated here technically "live" on a Lie group, in practice, all of our calculations are done at the Lie algebra level. Another more advanced source that covers the same material is ^[11].

As regarding precise definitions related to Lie algebras, we refer in the first instance to ^[12] and also to ^[11]. For a solvable Lie algebra, one should think roughly of a subspace of upper triangular matrices and for a nilpotent Lie algebra, a subspace of the strictly upper triangular matrices. Nonetheless, abelian sub-algebras are nilpotent, so subspaces of diagonal matrices are also nilpotent.

An important construct that we shall make use of is the semi-direct product of Lie algebras. The idea can be understood in various ways, but perhaps the simplest is to say that an algebra is a semi-direct product of Lie algebras if it is a vector space direct sum of a sub-algebra and an ideal. Solvable, not nilpotent, Lie algebras are only semi-direct products when there is an abelian complement to the nilradical. In this article, we shall be concerned with the Lie algebras $A_{6, 20}$ – $A_{6, 27}$ in ^[7]. Of these eight classes, only three, $A_{6, 22}, A_{6, 23}, A_{6, 27}$ for which $\epsilon = 0$ , are semi-direct products. However, we shall see the appearance of semi-direct products again when we analyze the symmetry algebras in Section 5. In general, a symmetry algebra need not be solvable, but rather will have a Levi decomposition, that is, it will be a semi-direct product of a solvable ideal (that itself may or may not be a semi-direct product) and a semi-simple sub-algebra. All of the algebras $A_{6, 20}$ – $A_{6, 27}$ studied in Section 5, produce semi-simple sub-algebras.

Concerning the definition of a linear connection, one may refer to ^[11] among a host of many excellent references. In relation to the current paper, one really only needs to understand that a linear connection produces a system of second order ordinary differential equations, the geodesics. These systems are similar to equations encountered in particle mechanics; the simplest example arises from the flat connection on Euclidean space (in arbitrary dimension), and the corresponding differential equations are the equations of motion of a free particle. More general connections introduce, as well as second order terms, first order terms that are quadratic in velocities.

Finally, we come to the notion of symmetry of a differential equation. Lie's original idea was that a differential equation that could be integrated explicitly must possess an underlying symmetry. By the term "symmetry", we understand a change of variables may be both independent and dependent variables, such that after applying a finite transformation, the differential equation remains invariant. For a determined system of ordinary differential equations, and later, partial differential equations, the set of such symmetries comprises what was to become known as a Lie transformation group. Very quickly it was realized that the underlying structure need not be associated to a differential equation at all, and led to the idea of an abstract Lie group. It was also understood by Lie and his contemporaries, that it would be virtually impossible to calculate Lie transformation groups explicitly, even in some of the simplest cases. That circumstance led Lie to another great insight: that it would be far easier to work at the infinitesimal level and find not the Lie group, but rather its Lie algebra. In fact, Lie frequently uses the term "group", whereas today we would be more careful and refer to the "Lie algebra".

In this work, Lie groups and Lie algebras appear at two levels. First of all, the differential equations that we study constitute an intrinsic part of the Lie group on which they are defined. Second, the set of symmetries of the differential equations itself forms a Lie group. However, the relationship between the two Lie groups and, more importantly, their associated Lie algebras is not a simple one in general. It is only in the case where the first Lie algebra has a trivial center that one can be sure that the first Lie algebra is isomorphic to a sub-algebra of the second; the sub-algebra in question is then either the algebra of left or right-invariant vector fields. In fact, the Lie algebras studied below in Section 5 all have a one-dimensional center.

3. The canonical connection on Lie groups

On left-invariant vector fields $X$ and $Y$ , the canonical symmetric connection $\nabla$ on a Lie group $G$ is defined by

$\begin{equation} \nabla_XY = \frac{1}{2} \ [X, Y], \end{equation}$

(3.1)

and then extended to arbitrary vector fields using linearity and the Leibnitz rule. The connection $\nabla$ is left-invariant. One could just as well use right-invariant vector fields to define $\nabla$ , but one must check that $\nabla$ is well-defined. Properties of the canonical connection have been studied in ^[11], and we will summarize them in the following proposition:

Proposition 1. For the canonical connection defined by (3.1):

(1) The torsion is zero.

(2) The curvature tensor $R$ is given by $R(X, Y)Z = \frac{1}{4}[[X, Y], Z]$ .

(3) The curvature tensor $R$ is covariantly constant.

(4) The curvature tensor $R$ is zero if, and only if, the Lie algebra is two-step nilpotent.

(5) The Ricci tensor is symmetric and in fact a multiple of the Killing form.

(6) The Ricci tensor is bi-invariant.

4. Symmetries of the geodesic equations

In this section, we explain the algorithm for finding the Lie symmetries of the geodesic equations. In local coordinates and in dimension $n$ , the geodesic equations are given by

$\begin{equation} \frac{d^2x^i}{dt^2}+\Gamma^i_{jk} \frac{dx^j}{dt}\frac{dx^k}{dt} = 0, \end{equation}$

(4.1)

where $\Gamma^i_{jk}$ are the connection components or Christoffel symbols, where $i, j, k = 1, ..., n$ . In dimension six, let's take our coordinates to be $t, p, q, x, y, z, w$ , where $t$ is the independent variable and $p, q, x, y, z, w$ are the dependant variables, so are functions of $t$ . Define $\Gamma$ to be

$\begin{equation} \Gamma = T \frac{\partial}{\partial t} + P \frac{\partial}{\partial p} + Q \frac{\partial}{\partial q} + X \frac{\partial}{\partial x} + Y \frac{\partial}{\partial y} + Z \frac{\partial}{\partial z} + W \frac{\partial}{\partial w}, \end{equation}$

(4.2)

where $T, P, Q, X, Y, Z$ , and $W$ are unknown functions of $(t, p, q, x, y, z, w)$ . The first prolongation $\Gamma^1$ and second prolongation $\Gamma^2$ of $\Gamma$ are given by

$\begin{align} \Gamma^{1} & = \Gamma + P_{t} \frac{\partial}{\partial\dot p} + Q_{t} \frac{\partial}{\partial \dot q} + X_{t} \frac{\partial}{\partial \dot x} + Y_{t} \frac{\partial}{\partial \dot y} + Z_{t} \frac{\partial}{\partial \dot z} + W_{t} \frac{\partial}{\partial \dot w}, \end{align}$

(4.3)

$\begin{align} \Gamma^{2} & = \Gamma^{1} + P_{tt} \frac{\partial}{\partial\ddot p} + Q_{tt} \frac{\partial}{\partial \ddot q} + X_{tt} \frac{\partial}{\partial \ddot x} + Y_{tt} \frac{\partial}{\partial \ddot y} + Z_{tt} \frac{\partial}{\partial \ddot z} + W_{tt} \frac{\partial}{\partial \ddot w}, \end{align}$

(4.4)

where

$\begin{align} \begin{array}{ll} & \quad P_{t} = D_{t}(P) - \dot {p} D_{t}(T), \quad \quad P_{tt} = D_{t}(P_{t}) - \ddot {p} D_{t}(T), \\ & \quad Q_{t} = D_{t}(Q) - \dot {q} D_{t}(T), \quad \quad Q_{tt} = D_{t}(Q_{t}) - \ddot {q} D_{t}(T) , \\ & \quad X_{t} = D_{t}(X) - \dot {x} D_{t}(T), \quad \quad X_{tt} = D_{t}(X_{t}) - \ddot {x} D_{t}(T) , \\ & \quad Y_{t} = D_{t}(Y) - \dot {y} D_{t}(T), \quad \quad Y_{tt} = D_{t}(Y_{t}) - \ddot {y} D_{t}(T) , \\ & \quad Z_{t} = D_{t}(Z) - \dot {z} D_{t}(T), \quad \quad Z_{tt} = D_{t}(Z_{t}) - \ddot {z} D_{t}(T), \\ &W_{t} = D_{t}(W) - \dot {w} D_{t}(T), \quad W_{tt} = D_{t}(W_{t}) - \ddot {w} D_{t}(T), \end{array} \end{align}$

(4.5)

and $D_{t}$ is given by

$\begin{equation} D_{t} = \frac{\partial}{\partial t} + \dot p \frac{\partial}{\partial p} + \dot q \frac{\partial}{\partial q} + \dot x \frac{\partial}{\partial x} + \dot y \frac{\partial}{\partial y} + \dot z \frac{\partial}{\partial z} + \dot w \frac{\partial}{\partial w} + \ddot p \frac{\partial}{\partial \dot p} + \ddot q \frac{\partial}{\partial \dot q} + \ddot x \frac{\partial}{\partial \dot x} + \ddot y \frac{\partial}{\partial \dot y} + \ddot z \frac{\partial}{\partial \dot z} + \ddot w \frac{\partial}{\partial \dot w}. \end{equation}$

(4.6)

Finally, $\Gamma$ is said to be a Lie symmetry of the system of the geodesic equations if

$\begin{equation} \Gamma^{2} (\Delta_{i}^{(2)}) |_{\Delta_{i}^{(2)} = 0} = 0, \end{equation}$

(4.7)

where

$\begin{equation} \Delta_{i}^{(2)} = \frac{d^2x^i}{dt^2}-f^{i} (t, x^{i}), \ \ \ \quad i = 1, 2, ..., 6. \end{equation}$

(4.8)

Equation (4.7) is called the Lie invariance condition. We equate the coefficients of the linearly independent derivation terms to zero, and this yields to an over-determined system of partial differential equations. For a good reference on symmetries of differential equations, we refer the reader to ^[13].

5. Lie symmetry algebras of $A_{6, 20}$ – $A_{6, 27}$

In this section, we consider the eight six-dimensional Lie algebras with co-dimension two nilradical and one-dimensional center, $A_{6, 20}$ – $A_{6, 27}$ in ^[7]. For each Lie algebra, we will list the nonzero brackets, the system of the geodesic equations and, the symmetry vector fields. Finally, we analyze the symmetry Lie algebra in terms of its nilradical, complement, and semi-simple sub-algebra.

5.1. Algebra $A^{ab}_{6, 20} (ab$ : $a^{2}+b^{2} \not = 0)$

The nonzero brackets for the algebra $A^{ab}_{6, 20}$ are given by

$\begin{equation} \begin{array}{lllll} [e_1, e_4] = ae_4, & [e_1, e_6] = e_6, & [e_2, e_4] = be_{4}, & [e_1, e_2] = e_{3}, & [e_2, e_5] = e_{5}. \end{array} \end{equation}$

(5.1)

The geodesic equations are given by

$\begin{equation} \begin{array}{llllll} \ddot{p} = \dot{p}( a \dot{z}+ b \dot{w} ), & \ddot{q} = \dot{q} \dot{z}, & \ddot{x} = \dot{x} \dot{w}, & \ddot{y} = \dot{z} \dot{w}, & \ddot{z} = 0, & \ddot{w} = 0.\\ \end{array} \end{equation}$

(5.2)

For the general case $A^{a \not = 0, b \not = 0}_{6, 20}$ , the symmetry Lie algebra is spanned by:

$\begin{equation} \begin{array}{lllll} e_{1} = D_{w}, & e_{2} = D_{z}, & e_{3} = t D_{t} , & e_{4} = D_{t}, & e_{5} = t D_{y}, \\ e_{6} = D_{p}, & e_{7} = D_{y}, & e_{8} = D_{q}, & e_{9} = D_{x}, & e_{10} = p D_{p}, \\ e_{11} = wD_{t}, & e_{12} = z D_{t}, & e_{13} = w D_{y}, & e_{14} = z D_{y}, & e_{15} = q D_{q}, \\ e_{16} = x D_{x}, & e_{17} = e^{z}D_{q}, & e_{18} = e^{w}D_{x}, & e_{19} = (w z-2y)D_{t}, & e_{20} = (w z-2y)D_{y}, \\ e_{21} = e^{bw}e^{az}D_{p}. \end{array} \end{equation}$

(5.3)

We make the following change of basis:

$\begin{equation} \begin{array}{lllll} \overline{e}_{1} = e_{4}, & \overline{e}_{2} = e_{6}, & \overline{e}_{3} = e_{7} , & \overline{e}_{4} = e_{8}, & \overline{e}_{5} = e_{9}, \\ \overline{e}_{6} = e_{11}, & \overline{e}_{7} = e_{12}, & \overline{e}_{8} = e_{13}, & \overline{e}_{9} = e_{14}, & \overline{e}_{10} = e_{17}, \\ \overline{e}_{11} = e_{18}, & \overline{e}_{12} = e_{21}, & \overline{e}_{13} = e_{1}+\frac{e_{14}}{2}, & \overline{e}_{14} = e_{2}+\frac{e_{13}}{2}, & \overline{e}_{15} = e_{3}-\frac{e_{20}}{2}, \\ \overline{e}_{16} = e_{10}, & \overline{e}_{17} = e_{15}, & \overline{e}_{18} = e_{16}, & \overline{e}_{19} = e_{3}+\frac{e_{20}}{2}, & \overline{e}_{20} = e_{5}, \\ \overline{e}_{21} = e_{19}. \end{array} \end{equation}$

(5.4)

The nonzero brackets of the symmetry algebra are given by

$\begin{equation} \begin{array}{llll} \left[e_{1} , e_{15}\right] = e_{1}, & \left[e_{1} , e_{19}\right] = e_1, & \left[e_{1}, e_{20}\right] = e_3, & \left[e_{2}, e_{16}\right] = e_{2}, \\ \left[e_{3}, e_{15}\right] = e_{3}, & \left[e_{3}, e_{19}\right] = - e_{3}, & \left[e_{3}, e_{21}\right] = -2 e_{1}, & \left[e_{4}, e_{17}\right] = e_{4}, \\ \left[e_{5}, e_{18}\right] = e_{5}, & \left[e_{6}, e_{13}\right] = - e_{1}, & \left[e_{6}, e_{15}\right] = e_{6}, & \left[e_{6}, e_{19}\right] = e_{6}, \\ \left[e_{6}, e_{20}\right] = e_{8}, & \left[e_{7}, e_{14}\right] = -e_{1}, & \left[e_{7}, e_{15}\right] = e_{7}, & \left[e_{7}, e_{19}\right] = e_{7}, \\ \left[e_{7}, e_{20}\right] = e_{9}, & \left[e_{8}, e_{13}\right] = - e_{3}, & \left[e_{8}, e_{15}\right] = e_{8}, & \left[e_{8}, e_{19}\right] = -e_{8}, \\ \left[e_{8}, e_{21}\right] = -2 e_{6}, & \left[e_{9}, e_{14}\right] = - e_{3}, & \left[e_{9}, e_{15}\right] = e_{9}, & \left[e_{9}, e_{19}\right] = -e_{9}, \\ \left[e_{9}, e_{21}\right] = -2 e_{7}, & \left[e_{10}, e_{14}\right] = - e_{10}, & \left[e_{10}, e_{17}\right] = e_{10}, & \left[e_{11}, e_{13}\right] = - e_{11}, \\ \left[e_{11}, e_{18}\right] = e_{11}, & \left[e_{12}, e_{13}\right] = -b e_{12}, & \left[e_{12}, e_{14}\right] = -a e_{12}, & \left[e_{12}, e_{16}\right] = e_{12}, \\ \left[e_{19}, e_{20}\right] = 2 e_{20}, & \left[e_{19}, e_{21}\right] = -2 e_{21}, & \left[e_{20}, e_{21}\right] = -2 e_{19}. \end{array} \end{equation}$

(5.5)

We describe the symmetry algebra by the following proposition:

Proposition 2. The symmetry Lie algebra is a twenty-one-dimensional Lie algebra. It is a semi-direct product of eighteen-dimensional solvable Lie algebra and $sl(2, \mbox{ $\Bbb R$ })$ . The solvable part is $(\mbox{ $\Bbb R$ }^{12} \rtimes \mbox{ $\Bbb R$ }^6)$ a semi-direct product of $\mbox{ $\Bbb R$ }^{12}$ and $\mbox{ $\Bbb R$ }^6$ . Therefore, the symmetry algebra can be identified as $(\mbox{ $\Bbb R$ }^{12} \rtimes \mbox{ $\Bbb R$ }^6) \rtimes sl(2, \mbox{ $\Bbb R$ })$ .

5.2. Algebra $A^{a}_{6, 21}$

The nonzero brackets for the algebra $A^{a}_{6, 21}$ are given by

$\begin{equation} \begin{array}{lllll} \left[e_1, e_4\right] = e_4, \quad \left[e_1, e_5\right] = e_6, \quad \left[e_2, e_4\right] = ae_{4}, \quad \left[e_2, e_5\right] = e_{5}, \quad \left[e_2, e_6\right] = e_{6}, \quad \left[e_1, e_2\right] = e_{3}. \end{array} \end{equation}$

(5.6)

The geodesic equations are given by

$\begin{equation} \begin{array}{llllll} \ddot{p} = \dot{p}( \dot{z}+ a \dot{w} ), & \ddot{q} = \dot{w}( \dot{q} - x\dot{z}) + \dot{z}\dot{x}, & \ddot{x} = \dot{x} \dot{w}, & \ddot{y} = \dot{z} \dot{w}, & \ddot{z} = 0, & \ddot{w} = 0.\\ \end{array} \end{equation}$

(5.7)

For the general case $A^{a \not = 0}_{6, 21}$ , the symmetry Lie algebra is spanned by

$\begin{equation} \begin{array}{llll} e_{1} = D_{t}, & e_{2} = tD_{y}, & e_{3} = D_{y} , & e_{4} = D_{p}, \\ e_{5} = D_{q}, & e_{6} = D_{z}, & e_{7} = D_{w}, & e_{8} = tD_{t}, \\ e_{9} = pD_{p}, & e_{10} = wD_{t}, & e_{11} = zD_{t}, & e_{12} = wD_{y}, \\ e_{13} = zD_{y}, & e_{14} = xD_{q}, & e_{15} = zD_{q}+D_{x}, & e_{16} = qD_{q}+xD_{x}, \\ e_{17} = e^{w}D_{q}, & e_{18} = e^{w}D_{x}, & e_{19} = (wz-2y)D_{t}, & e_{20} = (wz-2y)D_{y}, \\ e_{21} = e^{aw}e^{z}D_{p}. \end{array} \end{equation}$

(5.8)

We make the following change of basis:

$\begin{equation} \begin{array}{llllll} \overline{e}_{1} = e_{1}, & \overline{e}_{2} = e_{3}, & \overline{e}_{3} = e_{4} , & \overline{e}_{4} = e_{5}, & \overline{e}_{5} = e_{10}, & \overline{e}_{6} = e_{11}, \\ \overline{e}_{7} = e_{12}, & \overline{e}_{8} = e_{13}, & \overline{e}_{9} = e_{14}, & \overline{e}_{10} = e_{15}, & \overline{e}_{11} = e_{17}, & \overline{e}_{12} = e_{18}, \\ \overline{e}_{13} = e_{21}, & \overline{e}_{14} = e_{6}+\frac{e_{21}}{2}, & \overline{e}_{15} = e_{7}+\frac{e_{13}}{2}, & \overline{e}_{16} = e_{8}-\frac{e_{20}}{2}, & \overline{e}_{17} = e_{9}, & \overline{e}_{18} = e_{16}, \\ \overline{e}_{19} = e_{2}, & \overline{e}_{20} = e_{8}+\frac{e_{20}}{2}, & \overline{e}_{21} = e_{19}. \end{array} \end{equation}$

(5.9)

The nonzero brackets of the symmetry algebra are given by:

$\begin{equation} \begin{array}{llll} \left[e_{1} , e_{15}\right] = -e_{1}, & \left[e_{1} , e_{18}\right] = e_{1}, & \left[e_{2}, e_{18}\right] = e_{2}, & \left[e_{3}, e_{5}\right] = e_{1}, \\ \left[e_{3}, e_{15}\right] = -e_{3}, & \left[e_{3}, e_{18}\right] = e_{3}, & \left[e_{4}, e_{5}\right] = e_{2}, & \left[e_{4}, e_{14}\right] = -e_{2}, \\ \left[e_{4}, e_{18}\right] = e_{4}, & \left[e_{6}, e_{14}\right] = - e_{9}, & \left[e_{6}, e_{16}\right] = e_{6}, & \left[e_{6}, e_{19}\right] = e_{8}, \\ \left[e_{6}, e_{20}\right] = e_{6}, & \left[e_{7}, e_{15}\right] = -e_{10}, & \left[e_{7}, e_{16}\right] = e_{7}, & \left[e_{7}, e_{20}\right] = -e_{7}, \\ \left[e_{7}, e_{21}\right] = -2e_{12}, & \left[e_{8}, e_{14}\right] = - e_{10}, & \left[e_{8}, e_{16}\right] = e_{8}, & \left[e_{8}, e_{20}\right] = -e_{8}, \\ \left[e_{8}, e_{21}\right] = -2 e_{6}, & \left[e_{9}, e_{16}\right] = e_{9}, & \left[e_{9}, e_{19}\right] = e_{10}, & \left[e_{9}, e_{20}\right] = e_{9}, \\ \left[e_{10}, e_{16}\right] = e_{10}, & \left[e_{10}, e_{20}\right] = - e_{10}, & \left[e_{10}, e_{21}\right] = -2e_{9}, & \left[e_{11}, e_{17}\right] = e_{11}, \\ \left[e_{12}, e_{15}\right] = - e_{9}, & \left[e_{12}, e_{16}\right] = e_{12}, & \left[e_{12}, e_{19}\right] = e_{7}, & \left[e_{12}, e_{20}\right] = e_{12}, \\ \left[e_{13}, e_{14}\right] = - e_{13}, & \left[e_{13}, e_{15}\right] = -a e_{13}, & \left[e_{13}, e_{17}\right] = e_{13}, & \left[e_{19}, e_{20}\right] = -2 e_{19}, \\ \left[e_{19}, e_{21}\right] = -2 e_{20}, & \left[e_{20}, e_{21}\right] = -2 e_{21}. \end{array} \end{equation}$

(5.10)

We describe the symmetry algebra by the following proposition:

Proposition 3. The symmetry Lie algebra is a twenty-one-dimensional Lie algebra. It is a semi-direct product of eighteen-dimensional solvable Lie algebra and $sl(2, \mbox{ $\Bbb R$ })$ . The nilradical is thirteen-dimensional decomposable Lie algebra. In fact, the nilradical is a direct sum of $A_{5, 1}$ in Winternitz ^[5] and $\mbox{ $\Bbb R$ }^8$ . The nilradical has a five-dimensional abelian complement. Therefore, the symmetry algebra can be identified as

$( (A_{5, 1}\oplus \mathbb{R}^8) \rtimes \mathbb{R}^5) \rtimes sl(2, \mathbb{R}),$

where the nonzero brackets of $A_{5.1}$ are given by

$\begin{equation} [e_3, e_5] = e_1, \quad [e_4, e_5] = e_2. \end{equation}$

(5.11)

5.3. Algebra $A^{a\epsilon}_{6, 22} (a \epsilon$ : $a^{2}+\epsilon^{2}\not = 0, \epsilon= 0, 1)$

The nonzero brackets for the algebra $A^{a\epsilon}_{6, 22}$ are given by

$\begin{equation} \begin{array}{lllll} \left[e_1, e_3\right] = e_3, & \left[e_1, e_5\right] = e_6, & \left[e_2, e_4\right] = e_{4}, & \left[e_2, e_3\right] = a e_{3}, & \left[e_1, e_2\right] = \epsilon e_{5}. \end{array} \end{equation}$

(5.12)

5.3.1. $A^{\epsilon=0}_{6, 22}$

The geodesic equations are given by

$\begin{equation} \begin{array}{llllll} \ddot{p} = \dot{z}\dot{y} , & \ddot{q} = \dot{w} \dot{q}, & \ddot{x} = \dot{x} (\dot{z} + a\dot{w}), & \ddot{y} = 0, & \ddot{z} = 0 , & \ddot{w} = 0.\\ \end{array} \end{equation}$

(5.13)

For the general case $A^{a \not = 0, \epsilon = 0}_{6, 22}$ , the symmetry Lie algebra is spanned by

$\begin{equation} \begin{aligned} \begin{array}{llll} e_{1} = D_{y}, & e_{2} = D_{w}, & e_{3} = D_{z} , & e_{4} = tD_{t}, \\ e_{5} = D_{x}, & e_{6} = D_{t}, & e_{7} = t D_{p}, & e_{8} = D_{p}, \\ e_{9} = D_{q}, & e_{10} = x D_{x}, & e_{11} = w D_{t}, & e_{12} = y D_{t}, \\ e_{13} = z D_{t}, & e_{14} = w D_{p}, & e_{15} = y D_{p}, & e_{16} = z D_{p}, \\ e_{17} = q D_{q}, & e_{18} = p D_{p} +yD_{y}, & e_{19} = e^{w} D_{q}, & e_{20} = \frac{z^{2}}{2} D_{p} + z D_{y}, \\ e_{21} = \frac{zt}{2} D_{p} + t D_{y}, & e_{22} = (yz-2p) D_{t}, & e_{23} = (yz-2p) D_{p} , & e_{24} = \frac{wz}{2} D_{p} + w D_{y}, \\ e_{25} = e^{aw} e^{z} D_{x}, & e_{26} = (\frac{yz^{2}}{2} - pz )D_{p} + (yz-2p) D_{y}. \end{array}\end{aligned} \end{equation}$

(5.14)

We consider the following change of basis:

$\begin{equation} \begin{array}{lllll} \overline{e}_{1} = e_{1}, & \overline{e}_{2} = e_{5}, & \overline{e}_{3} = e_{6} , & \overline{e}_{4} = e_{8}, & \overline{e}_{5} = e_{9}, \\ \overline{e}_{6} = e_{11}, & \overline{e}_{7} = e_{13}, & \overline{e}_{8} = e_{14}, & \overline{e}_{9} = e_{16}, & \overline{e}_{10} = e_{19}, \\ \overline{e}_{11} = e_{20}, & \overline{e}_{12} = e_{24}, & \overline{e}_{13} = e_{25}, & \overline{e}_{14} = e_{2}, & \overline{e}_{15}\ = e_{3}+\frac{e_{15}}{2}, \\ \overline{e}_{16} = e_{4}+ e_{18}, & \overline{e}_{17} = e_{10}, & \overline{e}_{18} = e_{17}, & \overline{e}_{19} = e_{4} +\frac{e_{23}}{2}, & \overline{e}_{20} = e_{7}, \\ \overline{e}_{21} = e_{12}, & \overline{e}_{22} = e_{15}, & \overline{e}_{23} = e_{18}+ e_{23}, & \overline{e}_{24} = e_{21}, & \overline{e}_{25} = e_{22}, \\ \overline{e}_{26} = e_{26}. \end{array} \end{equation}$

(5.15)

The nonzero brackets of the symmetry algebra are given by

$\begin{equation} \begin{array}{lll} \left[e_{1} , e_{15}\right] = \frac{e_{4}}{2}, & \left[e_{1} , e_{16}\right] = e_{1}, & \left[e_{1}, e_{19}\right] = \frac{e_{9}}{2}, \\ \left[e_{1}, e_{21}\right] = e_{3}, & \left[e_{1}, e_{22}\right] = e_{4}, & \left[e_{1}, e_{23}\right] = e_{1} + e_{9}, \\ \left[e_{1}, e_{25}\right] = e_{7}, & \left[e_{1}, e_{26}\right] = e_{11}, & \left[e_{2}, e_{17}\right] = e_{2}, \\ \left[e_{3}, e_{16}\right] = e_{3}, & \left[e_{3}, e_{19}\right] = e_{3}, & \left[e_{3}, e_{20}\right] = e_{4}, \\ \left[e_{3}, e_{24}\right] = e_{1} + \frac{e_{9}}{2}, & \left[e_{4}, e_{16}\right] = e_{4}, & \left[e_{4}, e_{19}\right] = - e_{4}, \\ \left[e_{4}, e_{23}\right] = - e_{4}, & \left[e_{4}, e_{25}\right] = -2 e_{3}, & \left[e_{4}, e_{26}\right] = -2 e_{1} - e_{9}, \\ \left[e_{5}, e_{18}\right] = e_{5}, & \left[e_{6}, e_{14}\right] = -e_{3}, & \left[e_{6}, e_{16}\right] = e_{6}, \\ \left[e_{6}, e_{19}\right] = e_{6}, & \left[e_{6}, e_{20}\right] = e_{8}, & \left[e_{6}, e_{24}\right] = e_{12}, \\ \left[e_{7}, e_{15}\right] = - e_{3}, & \left[e_{7}, e_{16}\right] = e_{7}, & \left[e_{7}, e_{19}\right] = e_{7}, \\ \left[e_{7}, e_{20}\right] = e_{9}, & \left[e_{7}, e_{24}\right] = e_{11}, & \left[e_{8}, e_{14}\right] = - e_{4}, \\ \left[e_{8}, e_{16}\right] = e_{8}, & \left[e_{8}, e_{19}\right] = - e_{8}, & \left[e_{8}, e_{23}\right] = - e_{8}, \\ \left[e_{8}, e_{25}\right] = -2 e_{6}, & \left[e_{8}, e_{26}\right] = -2 e_{12}, & \left[e_{9}, e_{15}\right] = - e_{4}, \\ \left[e_{9}, e_{16}\right] = e_{9}, & \left[e_{9}, e_{19}\right] = - e_{9}, & \left[e_{9}, e_{23}\right] = - e_{9}, \\ \left[e_{9}, e_{25}\right] = -2 e_{7}, & \left[e_{9}, e_{26}\right] = -2 e_{11}, & \left[e_{10}, e_{14}\right] = - e_{10}, \\ \left[e_{10}, e_{18}\right] = e_{10}, & \left[e_{11}, e_{15}\right] = - e_{1} - \frac{9}{2}, & \left[e_{11}, e_{16}\right] = e_{11}, \\ \left[e_{11}, e_{21}\right] = e_{7}, & \left[e_{11}, e_{22}\right] = e_{9}, & \left[e_{11}, e_{23}\right] = e_{11}, \\ \left[e_{12}, e_{14}\right] = - e_{1} - \frac{9}{2}, & \left[e_{12}, e_{16}\right] = e_{12}, & \left[e_{12}, e_{21}\right] = e_{6}, \\ \left[e_{12}, e_{22}\right] = e_{8}, & \left[e_{12}, e_{23}\right] = e_{12}, & \left[e_{13}, e_{14}\right] = -a e_{13}, \\ \left[e_{13}, e_{15}\right] = - e_{13}, & \left[e_{13}, e_{17}\right] = e_{13}, & \left[e_{19}, e_{20}\right] = 2 e_{20}, \\ \left[e_{19}, e_{21}\right] = - e_{21}, & \left[e_{19}, e_{22}\right] = e_{22}, & \left[e_{19}, e_{24}\right] = e_{24}, \\ \left[e_{19}, e_{25}\right] = -2 e_{25}, & \left[e_{19}, e_{26}\right] = - e_{26}, & \left[e_{20}, e_{21}\right] = - e_{22}, \\ \left[e_{20}, e_{23}\right] = - e_{20}, & \left[e_{20}, e_{25}\right] = -2 e_{19}, & \left[e_{20}, e_{26}\right] = -2 e_{24}, \\ \left[e_{21}, e_{23}\right] = - e_{21}, & \left[e_{21}, e_{24}\right] = - e_{19} + e_{23}, & \left[e_{21}, e_{26}\right] = - e_{25}, \\ \left[e_{22}, e_{23}\right] = -2 e_{22}, & \left[e_{22}, e_{24}\right] = - e_{20}, & \left[e_{22}, e_{25}\right] = -2 e_{21}, \\ \left[e_{22}, e_{26}\right] = -2 e_{23}, & \left[e_{23}, e_{24}\right] = - e_{24}, & \left[e_{23}, e_{25}\right] = - e_{25}, \\ \left[e_{23}, e_{26}\right] = -2 e_{26}, & \left[e_{24}, e_{25}\right] = - e_{26}. \end{array} \end{equation}$

(5.16)

We describe the symmetry algebra by the following proposition:

Proposition 4. The symmetry Lie algebra is a twenty-six-dimensional Lie algebra. It is a semi-direct product of an eighteen-dimensional solvable Lie algebra and $sl(3, \mbox{ $\Bbb R$ })$ . The solvable part is $(\mbox{ $\Bbb R$ }^{13} \rtimes \mbox{ $\Bbb R$ }^5)$ , a semi-direct product of $\mbox{ $\Bbb R$ }^{13}$ and $\mbox{ $\Bbb R$ }^5$ . Therefore, the symmetry algebra can be identified as $(\mbox{ $\Bbb R$ }^{13} \rtimes \mbox{ $\Bbb R$ }^5) \rtimes sl(3, \mbox{ $\Bbb R$ })$ .

5.3.2. $A^{\epsilon=1}_{6, 22}$

The geodesic equations are given by

$\begin{equation} \begin{array}{llllll} \ddot{p} = \dot{p}(a \dot{z} + \dot{w}) , & \ddot{q} = \dot{q} \dot{z}, & \ddot{x} = \dot{y} \dot{w}, & \ddot{y} = \dot{z} \dot{w}, & \ddot{z} = 0 , & \ddot{w} = 0.\\ \end{array} \end{equation}$

(5.17)

For the general case $A^{a \not = 0, \epsilon = 1}_{6, 22}$ , the symmetry Lie algebra is spanned by

$\begin{equation} \begin{aligned} \begin{array}{llll} e_{1} = D_{t}, & e_{2} = t D_{x}, & e_{3} = D_{p} , & e_{4} = D_{x}, \\ e_{5} = D_{y}, & e_{6} = D_{q}, & e_{7} = D_{w}, & e_{8} = D_{z}, \\ e_{9} = t D_{t}, & e_{10} = p D_{p}, & e_{11} = w D_{t}, & e_{12} = z D_{t}, \\ e_{13} = z D_{x}, & e_{14} = w D_{x}, & e_{15} = q D_{q}, & e_{16} = y D_{x} + z D_{y}, \\ e_{17} = e^{z} D_{q}, & e_{18} = tw D_{x} + 2t D_{y}, & e_{19} = \frac{w^{2}}{2} D_{x} + w D_{y}, & e_{20} = wz D_{x} + 2z D_{y}, \\ e_{21} = (yz-2y) D_{t}, & e_{22} = e^{w} e^{az} D_{p}, & e_{23} = (wy-\frac{w^{2}z}{2}) D_{x} + (-wz+2y)D_{y}. \end{array}\end{aligned} \end{equation}$

(5.18)

We consider the following change of basis:

$\begin{equation} \begin{aligned} \begin{array}{lllll} \overline{e}_{1} = e_{1}, & \overline{e}_{2} = e_{2}, & \overline{e}_{3} = e_{3} , & \overline{e}_{4} = e_{4}, & \overline{e}_{5} = e_{5}, \\ \overline{e}_{6} = e_{6}, & \overline{e}_{7} = e_{11}, & \overline{e}_{8} = e_{12}, & \overline{e}_{9} = e_{13}, & \overline{e}_{10} = e_{14}, \\ \overline{e}_{11} = e_{16}, & \overline{e}_{12} = e_{17}, & \overline{e}_{13} = e_{19}, & \overline{e}_{14} = e_{20}, & \overline{e}_{15} = e_{22}, \\ \overline{e}_{16} = e_{7}, & \overline{e}_{17} = e_{8}, & \overline{e}_{18} = e_{9} +\frac{e_{23}}{2} , & \overline{e}_{19} = e_{10}, & \overline{e}_{20} = e_{15}, \\ \overline{e}_{21} = e_{9} - \frac{e_{23}}{2}, & \overline{e}_{22} = e_{18}, & \overline{e}_{23} = e_{21}. \end{array}\end{aligned} \end{equation}$

(5.19)

The nonzero brackets of the symmetry algebra are given by

$\begin{equation} \begin{aligned} \begin{array}{lll} \left[e_{1} , e_{2}\right] = e_{4}, & \left[e_{1} , e_{18}\right] = e_{1}, & \left[e_{1}, e_{21}\right] = e_{1}, \\ \left[e_{1}, e_{22}\right] = e_{10} + 2 e_{5}, & \left[e_{2}, e_{7}\right] = - e_{10}, & \left[e_{2}, e_{8}\right] = - e_{9}, \\ \left[e_{2}, e_{18}\right] = - e_{2}, & \left[e_{2}, e_{21}\right] = - e_{2}, & \left[e_{2}, e_{23}\right] = 2 e_{11} - e_{14}, \\ \left[e_{3}, e_{19}\right] = e_{3}, & \left[e_{5}, e_{11}\right] = e_{4}, & \left[e_{5}, e_{18}\right] = e_{5} + \frac{e_{10}}{2}, \\ \left[e_{5}, e_{21}\right] = - e_{5} - \frac{e_{10}}{2}, & \left[e_{5}, e_{23}\right] = -2 e_{1}, & \left[e_{6}, e_{20}\right] = e_{6}, \\ \left[e_{7}, e_{16}\right] = - e_{1}, & \left[e_{7}, e_{18}\right] = e_{7}, & \left[e_{7}, e_{21}\right] = e_{7}, \\ \left[e_{7}, e_{22}\right] = 2 e_{13}, & \left[e_{8}, e_{17}\right] = -e_{1}, & \left[e_{8}, e_{18}\right] = e_{8}, \\ \left[e_{8}, e_{21}\right] = e_{8}, & \left[e_{8}, e_{22}\right] = e_{14}, & \left[e_{9}, e_{17}\right] = - e_{4}, \\ \left[e_{10}, e_{16}\right] = - e_{4}, & \left[e_{11}, e_{13}\right] = - e_{10}, & \left[e_{11}, e_{14}\right] = - e_{9}, \\ \left[e_{11}, e_{17}\right] = - e_{5}, & \left[e_{11}, e_{18}\right] = - e_{11} + e_{14}, & \left[e_{11}, e_{21}\right] = e_{11} - e_{14}, \\ \left[e_{11}, e_{22}\right] = -2 e_{2}, & \left[e_{11}, e_{23}\right] = -2 e_{8}, & \left[e_{12}, e_{17}\right] = - e_{12}, \\ \left[e_{12}, e_{20}\right] = e_{12}, & \left[e_{13}, e_{16}\right] = - e_{10} - e_{5}, & \left[e_{13}, e_{18}\right] = e_{18}, \\ \left[e_{13}, e_{21}\right] = - e_{13}, & \left[e_{13}, e_{23}\right] = -2 e_{7}, & \left[e_{14}, e_{16}\right] = - e_{9}, \\ \left[e_{14}, e_{17}\right] = - e_{10} -2 e_{5}, & \left[e_{14}, e_{18}\right] = e_{14}, & \left[e_{14}, e_{21}\right] = - e_{14}, \\ \left[e_{14}, e_{23}\right] = -4 e_{8}, & \left[e_{15}, e_{16}\right] = - e_{15}, & \left[e_{15}, e_{17}\right] = -a e_{15}, \\ \left[e_{15}, e_{19}\right] = e_{15}, & \left[e_{16}, e_{18}\right] = \frac{e_{11}}{2} - \frac{e_{14}}{2}, & \left[e_{16}, e_{21}\right] = - \frac{e_{11}}{2} + \frac{e_{14}}{2}, \\ \left[e_{16}, e_{22}\right] = e_{2}, & \left[e_{16}, e_{23}\right] = e_{8}, & \left[e_{17}, e_{18}\right] = - \frac{e_{13}}{2}, \\ \left[e_{17}, e_{21}\right] = \frac{e_{13}}{2}, & \left[e_{17}, e_{23}\right] = e_{7}, & \left[e_{21}, e_{22}\right] = 2 e_{22}, \\ \left[e_{21}, e_{23}\right] = -2 e_{23}, & \left[e_{22}, e_{23}\right] = -4 e_{21}. \end{array}\end{aligned} \end{equation}$

(5.20)

We describe the symmetry algebra by the following proposition:

Proposition 5. The symmetry Lie algebra is a twenty-three-dimensional Lie algebra. It is a semi-direct product of twenty-dimensional solvable Lie algebra and $sl(2, \mbox{ $\Bbb R$ })$ . The solvable part is $(\mbox{ $\Bbb R$ }^{15} \rtimes \mbox{ $\Bbb R$ }^5)$ , a semi-direct product of $\mbox{ $\Bbb R$ }^{15}$ and $\mbox{ $\Bbb R$ }^5$ . Therefore, the symmetry algebra can be identified as $(\mbox{ $\Bbb R$ }^{15} \rtimes \mbox{ $\Bbb R$ }^5) \rtimes sl(2, \mbox{ $\Bbb R$ })$ .

5.4. Algebra $A^{a\epsilon}_{6, 23}(a \epsilon$ : $a \ge 0, \epsilon= 0, 1)$

The nonzero brackets for the algebra $A^{a\epsilon}_{6, 23}$ are given by

$\begin{equation} \begin{array}{llll} \left[e_1, e_3\right] = e_3, & \left[e_1, e_4\right] = e_4, & \left[e_1, e_5\right] = e_{6}, & \left[e_2, e_3\right] = e_{4}, \\ \left[e_2, e_4\right] = - e_{3}, & \left[e_2, e_5\right] = a e_{6}, & \left[e_1, e_2\right] = \epsilon e_{5}. \end{array} \end{equation}$

(5.21)

5.4.1. Case 1: $A^{a, \epsilon = 0}_{6, 23}$

The geodesic equations when $\epsilon = 0$ are given by

$\begin{equation} \begin{array}{llllll} \ddot{p} = \dot{p}\dot{z} - \dot{q}\dot{w} , & \ddot{q} = \dot{p}\dot{w} + \dot{q} \dot{z}, & \ddot{x} = \dot{y} (\dot{z} + a\dot{w}), & \ddot{y} = 0, & \ddot{z} = 0 , & \ddot{w} = 0.\\ \end{array} \end{equation}$

(5.22)

The symmetry Lie algebra is spanned by

$\begin{equation} \begin{array}{ll} e_{1} = D_{t}, & e_{2} = D_{p}, \\ e_{3} = D_{q} , & e_{4} = tD_{x}, \\ e_{5} = D_{x}, & e_{6} = D_{y}, \\ e_{7} = D_{w}, & e_{8} = D_{z}, \\ e_{9} = t D_{t}, & e_{10} = w D_{t}, \\ e_{11} = y D_{t}, & e_{12} = z D_{t}, \\ e_{13} = y D_{x}, & e_{14} = z D_{x}, \\ e_{15} = w D_{x}, & e_{16} = p D_{p} + q D_{q}, \\ e_{17} = x D_{x} + y D_{y}, & e_{18} = q D_{p} - p D_{q}, \\ e_{19} = \frac{t(aw+z)}{2} D_{x} + t D_{y}, & e_{20} = ((aw+z)y-2x) D_{x} , \\ e_{21} = \frac{w(aw+z)}{2} D_{x} + w D_{y}, & e_{22} = \frac{z(aw+z)}{2} D_{x} + z D_{y}, \\ e_{23} = \frac{((aw+z)y-2x)}{a} D_{t} , & e_{24} = \cos{(w)}e^{z} D_{p} + \sin{(w)}e^{z} D_{q}, \\ e_{25} = \sin{(w)}e^{z} D_{p} - \cos{(w)}e^{z} D_{q}, & e_{26} = \frac{(\frac{aw}{2}+\frac{z}{2})(awy+yz-2x)}{a} D_{x} + \frac{((aw+z)y-2x)}{a} D_{y}. \end{array} \end{equation}$

(5.23)

We consider the following change of basis:

$\begin{equation} \begin{array}{llllll} \overline{e}_{1} = e_{1}, & \overline{e}_{2} = e_{2}, & \overline{e}_{3} = e_{3} , & \overline{e}_{4} = e_{5}, & \overline{e}_{5} = e_{6}, & \overline{e}_{6} = e_{10}, \\ \overline{e}_{7} = e_{12}, & \overline{e}_{8} = e_{14}, & \overline{e}_{9} = e_{15}, & \overline{e}_{10} = e_{21}, & \overline{e}_{11} = e_{22}, & \overline{e}_{12} = e_{24}, \\ \overline{e}_{13} = e_{25}, & \overline{e}_{14} = e_{7} + \frac{ae_{13}}{2}, & \overline{e}_{15} = e_{8}+\frac{e_{13}}{2}, & \overline{e}_{16} = e_{9}+ e_{17}, & \overline{e}_{17} = e_{16}, & \overline{e}_{18} = e_{18}, \\ \overline{e}_{19} = e_{4} , & \overline{e}_{20} = e_{9} +\frac{e_{20}}{2}, & \overline{e}_{21} = e_{11}, & \overline{e}_{22} = e_{13}, & \overline{e}_{23} = e_{17}+ e_{20}, & \overline{e}_{24} = e_{19}, \\ \overline{e}_{25} = e_{23}, & \overline{e}_{26} = e_{26}. \end{array} \end{equation}$

(5.24)

The nonzero brackets of the symmetry algebra are given by

$\begin{equation} \begin{array}{lll} \left[e_{1} , e_{16}\right] = e_{1}, & \left[e_{1} , e_{19}\right] = e_{4}, & \left[e_{1}, e_{20}\right] = e_{1}, \\ \left[e_{1}, e_{24}\right] = \frac{ae_{9}}{2} + e_{5} + \frac{e_{8}}{2}, & \left[e_{2}, e_{17}\right] = e_{2}, & \left[e_{2}, e_{18}\right] = - e_{3}, \\ \left[e_{3}, e_{17}\right] = e_{3}, & \left[e_{3}, e_{18}\right] = e_{2}, & \left[e_{4}, e_{16}\right] = e_{4}, \\ \left[e_{4}, e_{20}\right] = - e_{4}, & \left[e_{4}, e_{23}\right] = - e_{4}, & \left[e_{4}, e_{25}\right] = - \frac{2e_{1}}{a} , \\ \left[e_{4}, e_{26}\right] = - e_{9} - \frac{e_{8}}{a} - \frac{2 e_{5}}{a}, & \left[e_{5}, e_{14}\right] = \frac{ae_{4}}{2}, & \left[e_{5}, e_{15}\right] = \frac{e_{4}}{2}, \\ \left[e_{5}, e_{16}\right] = e_{5}, & \left[e_{5}, e_{20}\right] = \frac{a e_{9}}{2} + \frac{e_{8}}{2}, & \left[e_{5}, e_{21}\right] = e_{1}, \\ \left[e_{5}, e_{22}\right] = e_{4}, & \left[e_{5}, e_{23}\right] = a e_{9} + e_{5} + e_{8}, & \left[e_{5}, e_{25}\right] = \frac{e_{7}}{a}+e_{6}, \\ \left[e_{5}, e_{26}\right] = \frac{e_{11}}{a}+e_{10}, & \left[e_{6}, e_{14}\right] = - e_{1}, & \left[e_{6}, e_{16}\right] = e_{6}, \\ \left[e_{6}, e_{19}\right] = e_{9}, & \left[e_{6}, e_{20}\right] = e_{6}, & \left[e_{6}, e_{24}\right] = e_{10}, \\ \left[e_{7}, e_{15}\right] = - e_{1}, & \left[e_{7}, e_{16}\right] = e_{7}, & \left[e_{7}, e_{19}\right] = e_{8}, \\ \left[e_{7}, e_{20}\right] = e_{7}, & \left[e_{7}, e_{24}\right] = e_{11}, & \left[e_{8}, e_{15}\right] = - e_{4}, \\ \left[e_{8}, e_{16}\right] = e_{8}, & \left[e_{8}, e_{20}\right] = - e_{8}, & \left[e_{8}, e_{23}\right] = - e_{8}, \\ \left[e_{8}, e_{25}\right] = - \frac{2e_{7}}{a}, & \left[e_{8}, e_{26}\right] = - \frac{2e_{11}}{a}, & \left[e_{9}, e_{14}\right] = - e_{4}, \\ \left[e_{9}, e_{16}\right] = e_{9}, & \left[e_{9}, e_{20}\right] = - e_{9}, & \left[e_{9}, e_{23}\right] = - e_{9}, \\ \left[e_{9}, e_{25}\right] = - \frac{2e_{6}}{a}, & \left[e_{9}, e_{26}\right] = - \frac{2e_{10}}{a}, & \left[e_{10}, e_{14}\right] = - \frac{a e_{9}}{2} - e_{5} - \frac{e_{8}}{2}, \\ \left[e_{10}, e_{16}\right] = e_{10}, & \left[e_{10}, e_{21}\right] = e_{6}, & \left[e_{10}, e_{22}\right] = e_{9}, \\ \left[e_{10}, e_{23}\right] = e_{10}, & \left[e_{11}, e_{15}\right] = - \frac{a e_{9}}{2} - e_{5} - \frac{e_{8}}{2}, & \left[e_{11}, e_{16}\right] = e_{11}, \\ \left[e_{11}, e_{21}\right] = e_{7}, & \left[e_{11}, e_{22}\right] = e_{8}, & \left[e_{11}, e_{23}\right] = e_{11}. \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\left[e_{12}, e_{14}\right] = e_{13}, & \left[e_{12}, e_{15}\right] = - e_{12}, & \left[e_{12}, e_{17}\right] = e_{12}, \\ \;\;\;\;\;\;\;\;\;\;\;\;\; \left[e_{12}, e_{18}\right] = e_{13}, & \left[e_{13}, e_{14}\right] = - e_{12}, & \left[e_{13}, e_{15}\right] = - e_{13}, \\\;\;\;\;\;\;\;\;\;\;\;\;\; \left[e_{13}, e_{17}\right] = e_{13}, & \left[e_{13}, e_{18}\right] = - e_{12}, & \left[e_{19}, e_{20}\right] = -2 e_{19}, \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\left[e_{19}, e_{21}\right] = - e_{22}, & \left[e_{19}, e_{23}\right] = - e_{19}, & \left[e_{19}, e_{25}\right] = - \frac{2 e_{20}}{a} , \\\;\;\;\;\;\;\;\;\;\;\;\;\; \left[e_{19}, e_{26}\right] = - \frac{2 e_{24}}{a} , & \left[e_{20}, e_{21}\right] = - e_{21}, & \left[e_{20}, e_{22}\right] = e_{22}, \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\left[e_{20}, e_{24}\right] = e_{24}, & \left[e_{20}, e_{25}\right] = -2 e_{25}, & \left[e_{20}, e_{26}\right] = - e_{26}, \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\left[e_{21}, e_{23}\right] = - e_{21}, & \left[e_{21}, e_{24}\right] = - e_{20} + e_{23}, & \left[e_{21}, e_{26}\right] = - e_{25}, \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\left[e_{22}, e_{23}\right] = -2 e_{22}, & \left[e_{22}, e_{24}\right] = - e_{19}, & \left[e_{22}, e_{25}\right] = - \frac{2 e_{21}}{a} , \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\left[e_{22}, e_{26}\right] = - \frac{2 e_{23}}{a} , & \left[e_{23}, e_{24}\right] = - e_{24}, & \left[e_{23}, e_{25}\right] = - e_{25}, \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\left[e_{23}, e_{26}\right] = -2 e_{26}, & \left[e_{24}, e_{25}\right] = - e_{26}. \end{array} \end{equation}$

(5.25)

We describe the symmetry algebra by the following proposition:

Proposition 6. The symmetry Lie algebra is a twenty-six- dimensional Lie algebra. It is a semi-direct product of eighteen-dimensional solvable Lie algebra and $sl(3, \mbox{ $\Bbb R$ })$ . The solvable part is $(\mbox{ $\Bbb R$ }^{13} \rtimes \mbox{ $\Bbb R$ }^5)$ , a semi-direct product of $\mbox{ $\Bbb R$ }^{13}$ and $\mbox{ $\Bbb R$ }^5$ . Therefore, the symmetry algebra can be identified as $(\mbox{ $\Bbb R$ }^{13} \rtimes \mbox{ $\Bbb R$ }^5) \rtimes sl(3, \mbox{ $\Bbb R$ })$ .

5.5. Case 2: $A^{a, \epsilon = 1}_{6, 23}$

5.5.1. $A^{a \not = 0, \epsilon = 1}_{6, 23}$

For $A^{a \not = 0, \epsilon = 1}_{6, 23}$ , the geodesic equations are given by

$\begin{equation} \begin{array}{llllll} \ddot{p} = \dot{p} \dot{z} + \dot{w}\dot{q} , & \ddot{q} = - \dot{p} \dot{w} +\dot{q} \dot{z}, & \ddot{x} = \dot{y}(\dot{z} +a \dot{w}), & \\ \ddot{y} = \dot{z}(\dot{z} +a \dot{w}), & \ddot{z} = 0 , & \ddot{w} = 0.\\ \end{array} \end{equation}$

(5.26)

The symmetry Lie algebra is spanned by

$\begin{equation} \begin{aligned} \begin{array}{lll} e_{1} = D_{w}, & e_{2} = D_{q}, & e_{3} = D_{p} , \\ e_{4} = D_{y}, & e_{5} = t D_{x}, & e_{6} = D_{z}, \\ e_{7} = D_{x}, & e_{8} = t D_{t}, & e_{9} = D_{t}, \\ e_{10} = w D_{x}, & e_{11} = z D_{x}, & e_{12} = w D_{t}, \\ e_{13} = z D_{t}, & e_{14} = y D_{x} + z D_{y}, & e_{15} = p D_{p} + q D_{q}, \\ e_{16} = -q D_{p} + p D_{q}, & e_{17}\ = \frac{t (aw+z)D_{x}}{2} + t D_{y}, & e_{18} = \frac{ (awz+z^{2}-2y)D_{x}}{a}, \\ e_{19} = \frac{ (awz+z^{2}-2y)D_{t}}{a} , & e_{20} = \frac{(\frac{a^{2} w^{2}}{2} - \frac{z^{2}}{2} + y)D_{x}}{a} + w D_{y}, & e_{21} = \sin(w) e^{z} D_{p} + \cos(w) e^{z} D_{q}, \\ e_{22} = - \cos(w) e^{z} D_{p} + \sin(w) e^{z} D_{q}, & e_{23} = \frac{(\frac{aw}{2} + \frac{z}{2}) (awz+ z^{2}- 2y)D_{x}}{a} + \frac{(awz+z^{2}-2y)D_{y}}{a}. \end{array}\end{aligned} \end{equation}$

(5.27)

We consider the following change of basis:

$\begin{equation} \begin{array}{lllll} \overline{e}_{1} = e_{4}, & \overline{e}_{2} = e_{5}, & \overline{e}_{3} = e_{7} , & \overline{e}_{4} = e_{9}, & \overline{e}_{5} = e_{10}, \\ \overline{e}_{6} = e_{11}, & \overline{e}_{7} = e_{12}, & \overline{e}_{8} = e_{13}, & \overline{e}_{9} = e_{14}, & \overline{e}_{10} = e_{18}, \\ \overline{e}_{11} = e_{20}, & \overline{e}_{12} = e_{2}, & \overline{e}_{13} = e_{3}, & \overline{e}_{14} = e_{21}, & \overline{e}_{15} = e_{22}, \\ \overline{e}_{16} = e_{1}, & \overline{e}_{17} = e_{6}, & \overline{e}_{18} = e_{8} -\frac{a e_{23}}{2} , & \overline{e}_{19} = e_{15}, & \overline{e}_{20} = e_{16}, \\ \overline{e}_{21} = e_{8} + \frac{a e_{23}}{2}, & \overline{e}_{22} = e_{17}, & \overline{e}_{23} = e_{19}. \end{array} \end{equation}$

(5.28)

The nonzero brackets of the symmetry algebra are given by

$\begin{equation} \begin{aligned} \begin{array}{ll} \left[e_{1} , e_{9}\right] = e_{3}, & \left[e_{1} , e_{10}\right] = - \frac{2 e_{3}}{a}, \\ \left[e_{1}, e_{11}\right] = \frac{e_{3}}{a}, & \left[e_{1}, e_{18}\right] = \frac{e_{6}}{2} + \frac{a e_{5}}{2} + e_{1}, \\ \left[e_{1}, e_{21}\right] = - \frac{e_{6}}{2} - \frac{a e_{5}}{2} - e_{1}, & \left[e_{1}, e_{23}\right] = - \frac{ 2 e_{4}}{2}, \\ \left[e_{2}, e_{4}\right] = - e_{3}, & \left[e_{2}, e_{7}\right] = - e_{5}, \\ \left[e_{2}, e_{8}\right] = - e_{6}, & \left[e_{2}, e_{18}\right] = - e_{2}, \\ \left[e_{2}, e_{21}\right] = - e_{2}, & \left[e_{2}, e_{23}\right] = - e_{10} , \\ \left[e_{4}, e_{18}\right] = e_{4} , & \left[e_{4}, e_{21}\right] = e_{4}, \\ \left[e_{4}, e_{22}\right] = \frac{e_{6}}{2} + \frac{a e_{5}}{2} + e_{1}, & \left[e_{5}, e_{16}\right] = - e_{3}, \\ \left[e_{6}, e_{17}\right] = - e_{3}, & \left[e_{7}, e_{16}\right] = - e_{4}, \\ \left[e_{7}, e_{18}\right] = e_{7}, & \left[e_{7}, e_{21}\right] = e_{7}, \\ \left[e_{7}, e_{22}\right] = \frac{e_{10}}{2} + e_{11}, & \left[e_{8}, e_{17}\right] = - e_{4}, \\ \left[e_{8}, e_{18}\right] = e_{8}, & \left[e_{8}, e_{21}\right] = e_{8}, \\ \left[e_{8}, e_{22}\right] = \frac{a e_{10}}{2} + e_{9}, & \left[e_{9}, e_{10}\right] = - \frac{2 e_{6}}{a}, \\ \left[e_{9}, e_{11}\right] = \frac{ e_{6}}{a} - e_{5}, & \left[e_{9}, e_{17}\right] = - e_{1}, \\ \left[e_{9}, e_{18}\right] = a e_{10} + e_{9}, & \left[e_{9}, e_{21}\right] = -a e_{10} - e_{9}, \\ \left[e_{9}, e_{22}\right] = - e_{2}, & \left[e_{9}, e_{23}\right] = - \frac{2 e_{8}}{a}, \\ \left[e_{10}, e_{11}\right] = \frac{2 e_{5}}{a}, & \left[e_{10}, e_{16}\right] = - e_{6}, \\ \left[e_{10}, e_{17}\right] = - \frac{2 e_{6}}{a} - e_{5}, & \left[e_{10}, e_{18}\right] = - e_{10}, \\ \left[e_{10}, e_{21}\right] = e_{10}, & \left[e_{10}, e_{22}\right] = \frac{2 e_{2}}{a}, \\ \left[e_{11}, e_{16}\right] = -a e_{5} - e_{1}, & \left[e_{11}, e_{17}\right] = \frac{e_{6}}{a}, \\ \left[e_{11}, e_{18}\right] = e_{10} + e_{11}, & \left[e_{11}, e_{21}\right] = - e_{10} - e_{11}, \\ \left[e_{11}, e_{22}\right] = -\frac{e_{2}}{a}, & \left[e_{11}, e_{23}\right] = - \frac{2 e_{7}}{a}, \\ \left[e_{12}, e_{19}\right] = e_{12}, & \left[e_{12}, e_{20}\right] = - e_{13}, \\ \left[e_{13}, e_{19}\right] = e_{13}, & \left[e_{13}, e_{20}\right] = e_{12}, \\ \left[e_{14}, e_{16}\right] = e_{15}, & \left[e_{14}, e_{17}\right] = - e_{14}, \\ \left[e_{14}, e_{19}\right] = e_{14}, & \left[e_{14}, e_{20}\right] = e_{15}. \end{array}\end{aligned} \end{equation}$

(5.29)

$\begin{equation} \begin{array}{ll} \left[e_{15}, e_{16}\right] = - e_{14}, & \left[e_{15}, e_{17}\right] = - e_{15}, \\ \left[e_{15}, e_{19}\right] = e_{15}, & \left[e_{15}, e_{20}\right] = - e_{14}, \\ \left[e_{16}, e_{18}\right] = - \frac{a^{2} e_{10}}{2} - \frac{a e_{9}}{2}, & \left[e_{16}, e_{21}\right] = \frac{a^{2} e_{10}}{2} + \frac{a e_{9}}{2}, \\ \left[e_{16}, e_{22}\right] = \frac{a e_{2}}{2}, & \left[e_{16}, e_{23}\right] = e_{8}\\ \left[e_{17}, e_{18}\right] = - \frac{a e_{11}}{2} - a e_{10} - e_{9}, & \left[e_{17}, e_{21}\right] = \frac{a e_{11}}{2}+ a e_{10} + e_{9}, \\ \left[e_{17}, e_{22}\right] = \frac{ e_{2}}{2}, & \left[e_{17}, e_{23}\right] = \frac{2 e_{8}}{a} + e_{7}, \\ \left[e_{21}, e_{22}\right] = 2 e_{22}, & \left[e_{21}, e_{23}\right] = -2 e_{23}, \\ \left[e_{22}, e_{23}\right] = -\frac{2 e_{21}}{a}. \end{array} \end{equation}$

(5.30)

We describe the symmetry algebra by the following proposition:

Proposition 7. The symmetry Lie algebra is a twenty-three- dimensional semi-direct product of twenty- dimensional solvable Lie algebra $S_{1, 20}$ and $sl(2, \mbox{ $\Bbb R$ })$ . The nilradical a fifteen-dimensional nilpotant Lie algebra $N_{1, 11} \oplus \mbox{ $\Bbb R$ }^4$ , which is a direct sum of $N_{1, 11}$ , an eleven-dimensional nilpotent Lie algebra, and a four-dimensional abelian Lie algebra $\mbox{ $\Bbb R$ }^4$ . The complement to the nilradical is a four-dimensional non-abelian. Therefore, the symmetry Lie algebra can be identified as $S_{1, 20}\rtimes sl(2, \mbox{ $\Bbb R$ })$ .

5.6. Algebra $A_{6, 24}$

The nonzero brackets for the algebra $A_{6, 24}$ are given by

$\begin{equation} \begin{array}{llll} \left[e_1, e_5\right] = e_5 + e_6, & \left[e_1, e_6\right] = e_6, & \left[e_2, e_4\right] = e_{4}, & \left[e_1, e_2\right] = e_{3}. \end{array} \end{equation}$

(5.31)

The geodesic equations are given by

$\begin{equation} \begin{array}{llllll} \ddot{p} = \dot{p}\dot{z}, & \ddot{q} = \dot{w}( \dot{q} + \dot{x}), & \ddot{x} = \dot{x} \dot{w}, & \ddot{y} = \dot{z} \dot{w}, & \ddot{z} = 0, & \ddot{w} = 0.\\ \end{array} \end{equation}$

(5.32)

The symmetry Lie algebra is spanned by

$\begin{equation} \begin{array}{lll} e_{1} = D_{t}, & e_{2} = tD_{y}, & e_{3} = D_{y} , \\ e_{4} = D_{p}, & e_{5} = D_{q}, & e_{6} = D_{x}, \\ e_{7} = D_{z}, & e_{8} = D_{w}, & e_{9} = t D_{t}, \\ e_{10} = p D_{p}, & e_{11} = w D_{t}, & e_{12} = z D_{t}, \\ e_{13} = w D_{y}, & e_{14} = z D_{y}, & e_{15} = x D_{q}, \\ e_{16} = q D_{q}+x D_{x}, & e_{17} = e^{w}D_{q}, & e_{18} = e^{z}D_{p}, \\ e_{19} = (wz-2y)D_{t}, & e_{20} = (wz-2y)D_{y}, & e_{21} = (w -1) e^{w} D_{q} + e^{w} D_{x}. \end{array} \end{equation}$

(5.33)

We consider the following change of basis:

$\begin{equation} \begin{array}{llll} \overline{e}_{1} = e_{5}, & \overline{e}_{2} = e_{17}, & \overline{e}_{3} = e_{6} , & \overline{e}_{4} = e_{21}, \\ \overline{e}_{5} = e_{15}, & \overline{e}_{6} = e_{1}, & \overline{e}_{7} = e_{3}, & \overline{e}_{8} = e_{4}, \\ \overline{e}_{9} = e_{11}, & \overline{e}_{10} = e_{12}, & \overline{e}_{11} = e_{13}, & \overline{e}_{12} = e_{14}, \\ \overline{e}_{13} = e_{18}, & \overline{e}_{14} = e_{7} +\frac{e_{13}}{2}, & \overline{e}_{15} = e_{8} +\frac{e_{14}}{2}, & \overline{e}_{16} = e_{9} -\frac{e_{20}}{2}, \\ \overline{e}_{17} = e_{10}, & \overline{e}_{18} = e_{16}, & \overline{e}_{19} = e_{2}, & \overline{e}_{20} = e_{9} +\frac{e_{20}}{2}, \\ \overline{e}_{21} = e_{19}, \end{array} \end{equation}$

(5.34)

and the nonzero brackets of the symmetry algebra are given by

$\begin{equation} \begin{array}{llll} \left[e_{1} , e_{18}\right] = e_{1}, & \left[e_{2} , e_{15}\right] = - e_{2}, & \left[e_{2}, e_{18}\right] = e_{2}, & \left[e_{3}, e_{5}\right] = e_{1}, \\ \left[e_{3}, e_{18}\right] = e_{3}, & \left[e_{4}, e_{5}\right] = e_{2}, & \left[e_{4}, e_{15}\right] = - e_{2} - e_{4}, & \left[e_{4}, e_{18}\right] = e_{4}, \\ \left[e_{6}, e_{16}\right] = e_{6}, & \left[e_{6}, e_{19}\right] = e_{7}, & \left[e_{6}, e_{20}\right] = e_{6}, & \left[e_{7}, e_{16}\right] = e_{7}, \\ \left[e_{7}, e_{20}\right] = - e_{7}, & \left[e_{7}, e_{21}\right] = -2 e_{6}, & \left[e_{8}, e_{17}\right] = e_{8}, & \left[e_{9}, e_{15}\right] = - e_{6}, \\ \left[e_{9}, e_{16}\right] = e_{9}, & \left[e_{9}, e_{19}\right] = e_{11}, & \left[e_{9}, e_{20}\right] = e_{9}, & \left[e_{10}, e_{14}\right] = - e_{6}, \\ \left[e_{10}, e_{16}\right] = e_{10}, & \left[e_{10}, e_{19}\right] = e_{12}, & \left[e_{10}, e_{20}\right] = e_{10}, & \left[e_{11}, e_{15}\right] = -e_{7}, \\ \left[e_{11}, e_{16}\right] = e_{11}, & \left[e_{11}, e_{20}\right] = - e_{11}, & \left[e_{11}, e_{21}\right] = -2 e_{9}, & \left[e_{12}, e_{14}\right] = - e_{7}, \\ \left[e_{12}, e_{16}\right] = e_{12}, & \left[e_{12}, e_{20}\right] = - e_{12}, & \left[e_{12}, e_{21}\right] = -2 e_{10}, & \left[e_{13}, e_{14}\right] = - e_{13}, \\ \left[e_{13}, e_{17}\right] = e_{13}, & \left[e_{19}, e_{20}\right] = -2 e_{19}, & \left[e_{19}, e_{21}\right] = -2 e_{20}, & \left[e_{20}, e_{21}\right] = -2 e_{21}. \end{array} \end{equation}$

(5.35)

We describe the symmetry algebra by the following proposition:

Proposition 8. The symmetry Lie algebra is a twenty-one-dimensional Lie algebra. It is a semi-direct product of an eighteen-dimensional solvable Lie algebra and $sl(2, \mbox{ $\Bbb R$ })$ . The nilradical is a thirteen-dimensional decomposable Lie algebra. In fact, the nilradical is a direct sum of $A_{5, 1}$ in Winternitz ^[5] and $\mbox{ $\Bbb R$ }^8$ . The nilradical has a five-dimensional abelian complement. Therefore, the symmetry algebra can be identified as $((A_{5, 1}\oplus \mbox{ $\Bbb R$ }^8) \rtimes \mbox{ $\Bbb R$ }^5) \rtimes sl(2, \mbox{ $\Bbb R$ })$ , where the nonzero brackets of $A_{5.1}$ are given by

$\begin{equation} [e_3, e_5] = e_1, \; \; \; \; [e_4, e_5] = e_2. \end{equation}$

(5.36)

5.7. Algebra $A^{ab}_{6, 25} (ab$ : $a^{2}+b^{2} \not = 0$ )

The nonzero brackets for the algebra $A^{ab}_{6, 25}$ are given by

$\begin{equation} \begin{array}{llll} \left[e_1, e_4\right] = ae_4, & \left[e_1, e_5\right] = e_6, & \left[e_1, e_6\right] = -e_{5}, & \left[e_2, e_4\right] = be_{4}, \\ \left[e_2, e_5\right] = e_{5}, & \left[e_2, e_6\right] = e_{6}, & \left[e_1, e_2\right] = e_{3}. \end{array} \end{equation}$

(5.37)

The geodesic equations are given by

$\begin{equation} \begin{array}{llllll} \ddot{p} = \dot{p}( b \dot{w}+ a \dot{z} ), & \ddot{q} = -\dot{w} \dot{z}, & \ddot{x} = \dot{x} \dot{z} - \dot{y} \dot{w}, & \ddot{y} = \dot{x} \dot{w} - \dot{y} \dot{z}, & \ddot{z} = 0, & \ddot{w} = 0.\\ \end{array} \end{equation}$

(5.38)

For the general case $A^{a \not = 0, b \not = 0}_{6, 25}$ , the symmetry Lie algebra is spanned by

$\begin{equation} \begin{array}{lll} e_{1} = D_{t}, & e_{2} = tD_{q}, & e_{3} = D_{q} , \\ e_{4} = D_{y}, & e_{5} = D_{x}, & e_{6} = D_{p}, \\ e_{7} = D_{w}, & e_{8} = D_{z}, & e_{9} = t D_{t}, \\ e_{10} = p D_{p}, & e_{11} = w D_{q}, & e_{12} = z D_{q}, \\ e_{13} = w D_{t}, & e_{14}\ = z D_{t}, & e_{15} = x D_{x} + y D_{y}, \\ e_{16} = (wz + 2q) D_{q}, & e_{17} = (wz + 2q) D_{t}, & e_{18} = e^{bw} e^{az} D_{p}. \end{array} \end{equation}$

(5.39)

We implement the following change of basis:

$\begin{equation} \begin{array}{llll} \overline{e}_{1} = e_{1}, & \overline{e}_{2} = e_{3}, & \overline{e}_{3} = e_{13} , & \overline{e}_{4} = e_{11}, \\ \overline{e}_{5} = - e_{7}+\frac{be_{8}}{a}, & \overline{e}_{6} = e_{4}, & \overline{e}_{7} = e_{5}, & \overline{e}_{8} = e_{6}, \\ \overline{e}_{9} = e_{11} + \frac{a e_{12}}{b}, & \overline{e}_{10} = e_{13}+\frac{a e_{14}}{b}, & \overline{e}_{11} = e_{18}, & \overline{e}_{12} = e_{8} -\frac{e_{11}}{2}, \\ \overline{e}_{13} = e_{9} +\frac{e_{16}}{2}, & \overline{e}_{14} = e_{10}, & \overline{e}_{15} = e_{15}, & \overline{e}_{16} = e_{2}, \\ \overline{e}_{17} = e_{9} -\frac{e_{16}}{2}, & \overline{e}_{18} = e_{17}, \end{array} \end{equation}$

(5.40)

and the nonzero brackets of the symmetry algebra are given by

$\begin{equation} \begin{array}{llll} \left[e_{1} , e_{13}\right] = e_{1}, & \left[e_{1} , e_{16}\right] = e_{2}, & \left[e_{1}, e_{17}\right] = e_{1}, \\ \left[e_{2}, e_{13}\right] = e_{2}, & \left[e_{2}, e_{17}\right] = - e_{2}, & \left[e_{2}, e_{18}\right] = 2 e_{1}, \\ \left[e_{3}, e_{5}\right] = e_{1}, & \left[e_{3}, e_{13}\right] = e_{3}, & \left[e_{3}, e_{16}\right] = e_{4}, \\ \left[e_{3}, e_{17}\right] = e_{3}, & \left[e_{4}, e_{5}\right] = e_{2}, & \left[e_{4}, e_{13}\right] = e_{4}, \\ \left[e_{4}, e_{17}\right] = - e_{4}, & \left[e_{4}, e_{18}\right] = 2 e_{3}, & \left[e_{5}, e_{12}\right] = \frac{e_{2}}{2}, \\ \left[e_{5}, e_{13}\right] = - \frac{b e_{9}}{2a} + \frac{b e_{4}}{a}, & \left[e_{5}, e_{17}\right] = \frac{b e_{9}}{2a} - \frac{b e_{4}}{a}, & \left[e_{5}, e_{18}\right] = - \frac{b e_{10}}{a} + \frac{2 b e_{3}}{a}, \\ \left[e_{6}, e_{15}\right] = e_{6}, & \left[e_{7}, e_{15}\right] = e_{7}, & \left[e_{8}, e_{14}\right] = e_{8}, \\ \left[e_{9}, e_{12}\right] = - \frac{a e_{2}}{b}, & \left[e_{9}, e_{13}\right] = e_{9}, & \left[e_{9}, e_{17}\right] = - e_{9}, \\ \left[e_{9}, e_{18}\right] = 2 e_{10}, & \left[e_{10}, e_{12}\right] = - \frac{a e_{1}}{b}, & \left[e_{10}, e_{13}\right] = e_{10}, \\ \left[e_{10}, e_{16}\right] = e_{9}, & \left[e_{10}, e_{17}\right] = e_{10}, & \left[e_{11}, e_{12}\right] = -a e_{11}, \\ \left[e_{11}, e_{14}\right] = e_{11}, & \left[e_{16}, e_{17}\right] = -2 e_{16}, & \left[e_{16}, e_{18}\right] = 2 e_{17}, \\ \left[e_{17}, e_{18}\right] = -2 e_{18}. \end{array} \end{equation}$

(5.41)

We describe the symmetry algebra by the following proposition:.

Proposition 9. The symmetry Lie algebra is an eighteen-dimensional Lie algebra. It is a semi-direct product of fifteen-dimensional solvable Lie algebra and $sl(2, \mbox{ $\Bbb R$ })$ . The nilradical is an eleven-dimensional decomposable Lie algebra. In fact, the nilradical is a direct sum of $A_{5, 1}$ in Winternitz ^[5] and $\mbox{ $\Bbb R$ }^6$ . The nilradical has a four-dimensional abelian complement. Therefore, the symmetry algebra can be identified as $((A_{5, 1}\oplus \mbox{ $\Bbb R$ }^4) \rtimes \mbox{ $\Bbb R$ }^5) \rtimes sl(2, \mbox{ $\Bbb R$ })$ , where the nonzero brackets of $A_{5.1}$ are given by

$\begin{equation} [e_3, e_5] = e_1, \; \; \; \; [e_4, e_5] = e_2. \end{equation}$

(5.42)

5.8. Algebra $A^{a}_{6, 26}$

The nonzero brackets for the algebra $A^{a}_{6, 26}$ are given by

$\begin{equation} \begin{array}{llll} \left[e_1, e_5\right] = ae_5 + e_{6}, & \left[e_1, e_6\right] = ae_6 - e_{5}, & \left[e_2, e_4\right] = e_{4}, & \left[e_1, e_2\right] = e_{3}. \end{array} \end{equation}$

(5.43)

The geodesic equations are given by

$\begin{equation} \begin{array}{llllll} \ddot{p} = \dot{p} \dot{z} , & \ddot{q} = \dot{w} \dot{z}, & \ddot{x} = \dot{w}(a \dot{x} - \dot{y} ), & \ddot{y} = \dot{w}( \dot{x} + a \dot{y} ), & \ddot{z} = 0, & \ddot{w} = 0.\\ \end{array} \end{equation}$

(5.44)

For the general case $A^{a \not = 0}_{6, 26}$ , the symmetry Lie algebra is spanned by

$\begin{equation} \begin{aligned} \begin{array}{llll} e_{1} = D_{t}, & e_{2} = t D_{q}, & e_{3} = D_{q} , & e_{4} = D_{x}, \\ e_{5} = D_{p}, & e_{6} = D_{y}, & e_{7} = D_{z}, & e_{8} = D_{w}, \\ e_{9} = t D_{t}, & e_{10} = p D_{p}, & e_{11} = w D_{q}, & e_{12} = z D_{q}, \\ e_{13} = w D_{t}, & e_{14} = z D_{t}, & e_{15} = x D_{x} + y D_{y}, & e_{16} = e^{z} D_{p}, \\ e_{17} = y D_{x} - x D_{y}, & e_{18} = (wz + 2q) D_{q}, & e_{19} = (wz + 2q) D_{t}, \\ e_{20} = e^{aw} \cos(w) D_{x} + e^{aw} \sin(w) D_{y}, & e_{21} = e^{aw} \sin(w) D_{x} - e^{aw} \cos(w) D_{y}. \end{array}\end{aligned} \end{equation}$

(5.45)

We consider the following change of basis:

$\begin{equation} \begin{array}{lllll} \overline{e}_{1} = e_{1}, & \overline{e}_{2} = e_{3}, & \overline{e}_{3} = e_{4} , & \overline{e}_{4} = e_{5}, & \overline{e}_{5} = e_{6}, \\ \overline{e}_{6} = e_{11}, & \overline{e}_{7} = e_{12}, & \overline{e}_{8} = e_{13}, & \overline{e}_{9} = e_{14}, & \overline{e}_{10} = e_{16}, \\ \overline{e}_{11} = e_{20}, & \overline{e}_{12} = e_{21}, & \overline{e}_{13} = e_{7} +\frac{e_{11}}{2}, & \overline{e}_{14} = e_{8} +\frac{e_{12}}{2}, & \overline{e}_{15} = e_{9} -\frac{e_{18}}{2}, \\ \overline{e}_{16} = e_{10}, & \overline{e}_{17} = e_{15}, & \overline{e}_{18} = e_{17}, & \overline{e}_{19} = e_{2}, & \overline{e}_{20} = e_{9} +\frac{e_{18}}{2}, \\ \overline{e}_{21} = e_{19}. \end{array} \end{equation}$

(5.46)

The nonzero brackets of the symmetry algebra are given by

$\begin{equation} \begin{array}{lll} \left[e_{1} , e_{15}\right] = e_{1}, & \left[e_{1} , e_{19}\right] = e_{2}, & \left[e_{1}, e_{20}\right] = e_{1}, \\ \left[e_{2}, e_{15}\right] = e_{2}, & \left[e_{2}, e_{20}\right] = - e_{2}, & \left[e_{2}, e_{21}\right] = -2 e_{1}, \\ \left[e_{3}, e_{17}\right] = e_{3}, & \left[e_{3}, e_{18}\right] = - e_{5}, & \left[e_{4}, e_{16}\right] = e_{4}, \\ \left[e_{5}, e_{17}\right] = e_{5}, & \left[e_{5}, e_{18}\right] = e_{3}, & \left[e_{6}, e_{14}\right] = - e_{2}, \\ \left[e_{6}, e_{15}\right] = e_{6}, & \left[e_{6}, e_{20}\right] = - e_{6}, & \left[e_{6}, e_{21}\right] = -2 e_{8}, \\ \left[e_{7}, e_{13}\right] = - e_{2}, & \left[e_{7}, e_{15}\right] = e_{7}, & \left[e_{7}, e_{20}\right] = - e_{7}, \\ \left[e_{7}, e_{21}\right] = -2 e_{9}, & \left[e_{8}, e_{14}\right] = - e_{1}, & \left[e_{8}, e_{15}\right] = e_{8}, \\ \left[e_{8}, e_{19}\right] = e_{6}, & \left[e_{8}, e_{20}\right] = e_{8}, & \left[e_{9}, e_{13}\right] = -e_{1}, \\ \left[e_{9}, e_{15}\right] = e_{9}, & \left[e_{9}, e_{19}\right] = e_{7}, & \left[e_{9}, e_{20}\right] = e_{9}, \\ \left[e_{10}, e_{13}\right] = - e_{10}, & \left[e_{10}, e_{16}\right] = e_{10}, & \left[e_{11}, e_{14}\right] = -a e_{11} + e_{12}, \\ \left[e_{11}, e_{17}\right] = e_{11}, & \left[e_{11}, e_{18}\right] = e_{12}, & \left[e_{12}, e_{14}\right] = -a e_{12} - e_{11}, \\ \left[e_{12}, e_{17}\right] = e_{12}, & \left[e_{12}, e_{18}\right] = - e_{11}, & \left[e_{19}, e_{20}\right] = -2 e_{19}, \\ \left[e_{19}, e_{21}\right] = -2 e_{20}, & \left[e_{20}, e_{21}\right] = -2 e_{21}. \end{array} \end{equation}$

(5.47)

We describe the symmetry algebra by the following proposition:

Proposition 10. The symmetry Lie algebra is a twenty-one-dimensional Lie algebra. It is a semi-direct product of an eighteen-dimensional solvable Lie algebra and $sl(2, \mbox{ $\Bbb R$ })$ . The nilradical is twelve-dimensional abelian Lie algebra and has a six-dimensional abelian complement. Therefore, the symmetry algebra can be identified as: $(\mbox{ $\Bbb R$ }^{12} \rtimes \mbox{ $\Bbb R$ }^{6}) \rtimes sl(2, \mbox{ $\Bbb R$ })$ .

5.8.1. Case 1: $A^{a = 0}_{6, 26}$

The symmetry Lie algebra is spanned by

$\begin{equation} \begin{aligned} \begin{array}{lll} e_{1} = D_{t}, & e_{2} = t D_{q}, & e_{3} = D_{q} , \\ e_{4} = D_{p}, & e_{5} = D_{x}, & e_{6} = D_{y}, \\ e_{7} = D_{z}, & e_{8} = D_{w}, & e_{9} = t D_{t}, \\ e_{10} = p D_{p}, & e_{11} = w D_{q}, & e_{12} = z D_{p}, \\ e_{13} = w D_{t}, & e_{14} = z D_{t} , & e_{15} = x D_{x} + y D_{y}, \\ e_{16} = e^{z} D_{p}, & e_{17} = y D_{x} - x D_{y}, & e_{18} = \cos(w) D_{x} + \sin(w) D_{y}, \\ e_{19} = (wz - 2q) D_{q} , & e_{20} = (wz - 2q) D_{t}, & e_{21} = \sin(w) D_{x} - \cos(w) D_{y}, \\ e_{22} = ( - \cos(w) y + x \sin(w)) D_{x} + (- \cos(w) x - y \sin(w)) D_{y}, & \\ e_{23} = ( \cos(w) x + y \sin(w)) D_{x} + (- \cos(w) y + x \sin(w)) D_{y}.& \\ \end{array}\end{aligned} \end{equation}$

(5.48)

We implement the following change of basis

$\begin{equation} \begin{array}{llll} \overline{e}_{1} = e_{1}, & \overline{e}_{2} = e_{3}, & \overline{e}_{3} = e_{4} , & \overline{e}_{4} = e_{5}, \\ \overline{e}_{5} = e_{6}, & \overline{e}_{6} = e_{11}, & \overline{e}_{7} = e_{12}, & \overline{e}_{8} = e_{13}, \\ \overline{e}_{9} = e_{14}, & \overline{e}_{10} = e_{16}, & \overline{e}_{11} = e_{18}, & \overline{e}_{12} = e_{21}, \\ \overline{e}_{13} = e_{7} + \frac{e_{11}}{2}, & \overline{e}_{14} = e_{8} -\frac{e_{17}}{2} + \frac{e_{12}}{2}, & \overline{e}_{15} = e_{9} - \frac{e_{19}}{2}, & \overline{e}_{16} = e_{10}, \\ \overline{e}_{17} = e_{15}, & \overline{e}_{18} = e_{2}, & \overline{e}_{19} = e_{9} + \frac{e_{19}}{2}, & \overline{e}_{20} = e_{20}, \\ \overline{e}_{21} = e_{17}, & \overline{e}_{22} = e_{22}, & \overline{e}_{23} = e_{23}, \end{array} \end{equation}$

(5.49)

and the nonzero brackets of the symmetry algebra are given by

$\begin{equation} \begin{aligned} \begin{array}{llll} \left[e_{1} , e_{15}\right] = e_{1}, & \left[e_{1} , e_{18}\right] = e_{2}, & \left[e_{1}, e_{19}\right] = - e_{1} , & \left[e_{2}, e_{15}\right] = e_{2}, \\ \left[e_{2}, e_{19}\right] = - e_{2}, & \left[e_{2}, e_{20}\right] = -2 e_{1}, & \left[e_{3}, e_{16}\right] = e_{3}, & \left[e_{4}, e_{14}\right] = \frac{ e_{5}}{2}, \\ \left[e_{4}, e_{17}\right] = e_{4}, & \left[e_{4}, e_{21}\right] = - e_{5}, & \left[e_{4}, e_{22}\right] = e_{12}, & \left[e_{4}, e_{23}\right] = e_{11} , \\ \left[e_{5}, e_{14}\right] = - \frac{ e_{4}}{2} , & \left[e_{5}, e_{17}\right] = e_{5}, & \left[e_{5}, e_{21}\right] = e_{4}, & \left[e_{5}, e_{22}\right] = - e_{11}, \\ \left[e_{5}, e_{23}\right] = e_{12}, & \left[e_{6}, e_{14}\right] = - e_{2}, & \left[e_{6}, e_{15}\right] = e_{6}, & \left[e_{6}, e_{19}\right] = - e_{6}, \\ \left[e_{6}, e_{20}\right] = -2 e_{8}, & \left[e_{7}, e_{13}\right] = - e_{2}, & \left[e_{7}, e_{15}\right] = e_{7}, & \left[e_{7}, e_{19}\right] = - e_{7}, \\ \left[e_{7}, e_{20}\right] = -2 e_{9}, & \left[e_{8}, e_{14}\right] = - e_{1}, & \left[e_{8}, e_{15}\right] = e_{8}, & \left[e_{8}, e_{18}\right] = e_{6}, \\ \left[e_{8}, e_{19}\right] = e_{8}, & \left[e_{9}, e_{13}\right] = - e_{1}, & \left[e_{9}, e_{15}\right] = e_{9} , & \left[e_{9}, e_{18}\right] = e_{7}, \\ \left[e_{9}, e_{19}\right] = e_{9}, & \left[e_{10}, e_{13}\right] = - e_{10}, & \left[e_{10}, e_{16}\right] = e_{10} , & \left[e_{11}, e_{14}\right] = \frac{ e_{12}}{2}, \\ \left[e_{11}, e_{17}\right] = e_{11}, & \left[e_{11}, e_{21}\right] = e_{12}, & \left[e_{11}, e_{22}\right] = - e_{5}, & \left[e_{11}, e_{23}\right] = e_{4}, \\ \left[e_{12}, e_{14}\right] = - \frac{ e_{11}}{2}, & \left[e_{12}, e_{17}\right] = e_{12}, & \left[e_{12}, e_{21}\right] = - e_{11}, & \left[e_{12}, e_{22}\right] = e_{4}, \\ \left[e_{12}, e_{23}\right] = e_{5}, & \left[e_{18}, e_{19}\right] = - e_{18}, & \left[e_{18}, e_{20}\right] = -2 e_{19}, & \left[e_{19}, e_{20}\right] = -2 e_{20}, \\ \left[e_{21}, e_{22}\right] = 2 e_{23}, & \left[e_{21}, e_{23}\right] = -2 e_{22}, & \left[e_{22}, e_{23}\right] = -2 e_{21}. \end{array}\end{aligned} \end{equation}$

(5.50)

We describe the symmetry algebra by the following proposition:

Proposition 11. The symmetry Lie algebra is a twenty-three-dimensional Lie algebra. It is a semi-direct product of a seventeen-dimensional solvable Lie algebra and two copies of $sl(2, \mbox{ $\Bbb R$ })$ . Furthermore, the symmetry Lie algebra has a twelve-dimensional abelian nilradical and five-dimensional abelian complement. Therefore, the symmetry algebra can be identified as

$( \mathbb{R}^{12} \rtimes \mathbb{R}^{5}) \rtimes (sl(2, \mathbb{R}) \oplus sl(2, \mathbb{R})).$

5.9. Algebra $A^{\epsilon}_{6, 27}$ : $(\epsilon = 0, 1$ )

The nonzero brackets for the algebra $A^{\epsilon}_{6, 27}$ are given by

$\begin{equation} \left[e_1, e_3\right] = e_4 , \; \; \; \left[e_1, e_5\right] = e_6 , \; \; \; \left[e_1, e_6\right] = - e_{5}, \; \; \; \left[e_2, e_5\right] = e_{5}, \; \; \; \left[e_2, e_6\right] = e_{6}, \; \; \; \left[e_1, e_2\right] = \epsilon e_{3}. \end{equation}$

(5.51)

5.9.1. $A^{\epsilon=0}_{6, 27}$

The geodesic equations where $\epsilon = 0$ are given by

$\begin{equation} \begin{array}{llllll} \ddot{p} = \dot{p} \dot{w} - \dot{q} \dot{z} , & \ddot{q} = \dot{p} \dot{z} + \dot{q} \dot{w}, & \ddot{x} = 0, & \ddot{y} = \dot{x} \dot{z}, & \ddot{z} = 0, & \ddot{w} = 0.\\ \end{array} \end{equation}$

(5.52)

The symmetry Lie algebra is spanned by

$\begin{equation} \begin{aligned} \begin{array}{ll} e_{1} = D_{t}, & e_{2} = t D_{y}, \\ e_{3} = D_{y} , & e_{4} = D_{p}, \\ e_{5} = D_{q}, & e_{6} = D_{x}, \\ e_{7} = D_{w}, & e_{8} = D_{z}, \\ e_{9} = t D_{t}, & e_{10} = w D_{t}, \\ e_{11} = x D_{t}, & e_{12} = z D_{t}, \\ e_{13} = w D_{y}, & e_{14} = x D_{y}, \\ e_{15} = z D_{y}, & e_{16} = p D_{p} + q D_{q}, \\ e_{17} = x D_{x} + y D_{y}, & e_{18} = q D_{p} - p D_{q}, \\ e_{19} = t D_{x} + \frac{tz}{2} D_{y}, & e_{20} = z D_{x} + \frac{z^2}{2} D_{y} , \\ e_{21} = (xz - 2y) D_{t}, & e_{22}\ = (xz - 2y) D_{y}, \\ e_{23} = w D_{x} + \frac{wz}{2} D_{y} , & e_{24} = e^{w} \cos(z) D_{p} + e^{w} \sin(z) D_{q}, \\ e_{25} = e^{w} \sin(z) D_{p}- e^{w} \cos(z) D_{q}, & e_{26} = (xz- 2y )D_{x} + (\frac{x z^2}{2} - yz) D_{y}. \end{array}\end{aligned} \end{equation}$

(5.53)

We implement the following change of basis:

$\begin{equation} \begin{aligned} \begin{array}{llll} \overline{e}_{1} = e_{1}, & \overline{e}_{2} = e_{3}, & \overline{e}_{3} = e_{4} , & \overline{e}_{4} = e_{5}, \\ \overline{e}_{5} = e_{6}, & \overline{e}_{6} = e_{10}, & \overline{e}_{7} = e_{12}, & \overline{e}_{8} = e_{13}, \\ \overline{e}_{9} = e_{15}, & \overline{e}_{10} = e_{20}, & \overline{e}_{11} = e_{23}, & \overline{e}_{12} = e_{24}, \\ \overline{e}_{13} = e_{25}, & \overline{e}_{14} = e_{7} , & \overline{e}_{15} = e_{8} +\frac{e_{14}}{2}, & \overline{e}_{16} = e_{9}+ e_{17}, \\ \overline{e}_{17} = e_{16}, & \overline{e}_{18} = e_{18}, & \overline{e}_{19} = e_{2} , & \overline{e}_{20} = e_{9} +\frac{e_{22}}{2}, \\ \overline{e}_{21} = e_{11}, & \overline{e}_{22} = e_{14}, & \overline{e}_{23} = e_{17}+ e_{22}, & \overline{e}_{24} = e_{19}, \\ \overline{e}_{25} = e_{21}, & \overline{e}_{26} = e_{26}, \end{array}\end{aligned} \end{equation}$

(5.54)

and the nonzero brackets of the symmetry algebra are given by

$\begin{equation} \begin{aligned} \begin{array}{lll} \left[e_{1} , e_{16}\right] = e_{1}, & \left[e_{1} , e_{19}\right] = e_{2}, & \left[e_{1}, e_{20}\right] = e_{1}, \\ \left[e_{1}, e_{24}\right] = e_{5} +\frac{e_{9}}{2}, & \left[e_{2}, e_{16}\right] = e_{2}, & \left[e_{2}, e_{20}\right] = - e_{2}, \\ \left[e_{2}, e_{23}\right] = - e_{2}, & \left[e_{2}, e_{25}\right] = -2 e_{1}, & \left[e_{2}, e_{26}\right] = -2 e_{5} - e_{9}, \\ \left[e_{3}, e_{17}\right] = e_{3}, & \left[e_{3}, e_{18}\right] = - e_{4}, & \left[e_{4}, e_{17}\right] = e_{4} , \\ \left[e_{4}, e_{18}\right] = e_{3}, & \left[e_{5}, e_{15}\right] = \frac{e_{2}}{2}, & \left[e_{5}, e_{16}\right] = e_{5}, \\ \left[e_{5}, e_{20}\right] = \frac{e_{9}}{2}, & \left[e_{5}, e_{21}\right] = e_{1}, & \left[e_{5}, e_{22}\right] = e_{2}, \\ \left[e_{5}, e_{23}\right] = e_{5} + e_{9}, & \left[e_{5}, e_{25}\right] = e_{7} , & \left[e_{5}, e_{26}\right] = e_{10}, \\ \left[e_{6}, e_{14}\right] = - e_{1}, & \left[e_{6}, e_{16}\right] = e_{6}, & \left[e_{6}, e_{19}\right] = e_{8}, \\ \left[e_{6}, e_{20}\right] = e_{6}, & \left[e_{6}, e_{24}\right] = e_{11}, & \left[e_{7}, e_{15}\right] = - e_{1}, \\ \left[e_{7}, e_{16}\right] = e_{7}, & \left[e_{7}, e_{19}\right] = e_{9}, & \left[e_{7}, e_{20}\right] = e_{7}, \\ \left[e_{7}, e_{24}\right] = e_{10}, & \left[e_{8}, e_{14}\right] = - e_{2}, & \left[e_{8}, e_{16}\right] = e_{8}, \\ \left[e_{8}, e_{20}\right] = - e_{8}, & \left[e_{8}, e_{23}\right] = - e_{8}, & \left[e_{8}, e_{25}\right] = -2 e_{6}, \\ \left[e_{8}, e_{26}\right] = -2 e_{11}, & \left[e_{9}, e_{15}\right] = - e_{2}, & \left[e_{9}, e_{16}\right] = e_{9}, \\ \left[e_{9}, e_{20}\right] = - e_{9}, & \left[e_{9}, e_{23}\right] = - e_{9}, & \left[e_{9}, e_{25}\right] = -2 e_{7}, \\ \left[e_{9}, e_{26}\right] = -2 e_{10}, & \left[e_{10}, e_{15}\right] = - e_{5} - \frac{e_{9}}{2}, & \left[e_{10}, e_{16}\right] = e_{10} , \\ \left[e_{10}, e_{21}\right] = e_{7}, & \left[e_{10}, e_{22}\right] = e_{9}, & \left[e_{10}, e_{23}\right] = e_{10}, \\ \left[e_{11}, e_{14}\right] = - e_{5} - \frac{e_{9}}{2}, & \left[e_{11}, e_{16}\right] = e_{11}, & \left[e_{11}, e_{21}\right] = e_{6}, \\ \left[e_{11}, e_{22}\right] = e_{8}, & \left[e_{11}, e_{23}\right] = e_{11}, & \left[e_{12}, e_{14}\right] = - e_{12}, \\ \left[e_{12}, e_{15}\right] = e_{13}, & \left[e_{12}, e_{17}\right] = e_{12}, & \left[e_{12}, e_{18}\right] = e_{13}, \\ \left[e_{13}, e_{14}\right] = - e_{13}, & \left[e_{13}, e_{15}\right] = - e_{12}, & \left[e_{13}, e_{17}\right] = e_{13}, \\ \left[e_{13}, e_{18}\right] = - e_{12}, & \left[e_{19}, e_{20}\right] = -2 e_{19}, & \left[e_{19}, e_{21}\right] = - e_{22}, \\ \left[e_{19}, e_{23}\right] = - e_{19}, & \left[e_{19}, e_{25}\right] = -2 e_{20}, & \left[e_{19}, e_{26}\right] = -2 e_{24} , \\ \left[e_{20}, e_{21}\right] = - e_{21} , & \left[e_{20}, e_{22}\right] = e_{22}, & \left[e_{20}, e_{24}\right] = e_{24}, \\ \left[e_{20}, e_{25}\right] = -2 e_{25}, & \left[e_{20}, e_{26}\right] = - e_{26}, & \left[e_{21}, e_{23}\right] = - e_{21}, \\ \left[e_{21}, e_{24}\right] = - e_{20} + e_{23}, & \left[e_{21}, e_{26}\right] = - e_{25} , & \left[e_{22}, e_{23}\right] = -2 e_{22}, \\ \left[e_{22}, e_{24}\right] = - e_{19}, & \left[e_{22}, e_{25}\right] = -2 e_{21}, & \left[e_{22}, e_{26}\right] = -2 e_{23} , \\ \left[e_{23}, e_{24}\right] = - e_{24} , & \left[e_{23}, e_{25}\right] = - e_{25}, & \left[e_{23}, e_{26}\right] = -2 e_{26}, \\ \left[e_{24}, e_{25}\right] = - e_{26}. \end{array}\end{aligned} \end{equation}$

(5.55)

We describe the symmetry algebra by the following proposition:

Proposition 12. The symmetry Lie algebra is a twenty-six-dimensional semi-direct product of an eighteen solvable Lie algebra and eight-dimensional semi-simple $sl(3, \mbox{ $\Bbb R$ })$ . Furthermore, the symmetry Lie algebra has a thirteen-dimensional abelian nilradical. Therefore, the symmetry algebra can be identified as: $(\mbox{ $\Bbb R$ }^{13} \rtimes \mbox{ $\Bbb R$ }^5) \rtimes sl(3, \mbox{ $\Bbb R$ })$ .

5.9.2. $A^{\epsilon=1}_{6, 27}$

The geodesic equations where $\epsilon = 1$ are given by

$\begin{equation} \ddot{p} = \dot{q} \dot{w} , \quad \ddot{q} = \dot{z} \dot{w}, \quad \ddot{x} = \dot{z} \dot{x} - \dot{w} \dot{y}, \quad \ddot{y} = \dot{z} \dot{y} + \dot{w} \dot{x}, \quad \ddot{z} = 0, \quad\ddot{w} = 0. \end{equation}$

(5.56)

The symmetry Lie algebra is spanned by

$\begin{equation} \begin{aligned} \begin{array}{llll} e_{1} = D_{z}, & e_{2} = D_{p}, & e_{3} = D_{x} , & e_{4} = D_{w}, \\ e_{5} = D_{y}, & e_{6} = D_{q}, & e_{7} = t D_{t}, & e_{8} = D_{t}, \\ e_{9} = t D_{p}, & e_{10} = z D_{p}, & e_{11} = w D_{p}, & e_{12} = w D_{t}, \\ e_{13} = z D_{t}, & e_{14} = q D_{p} + z D_{q}, & e_{15} = x D_{x} + y D_{y}, & e_{16} = y D_{x} - x D_{y}, \\ e_{17} = tw D_{p} + 2t D_{q}, & e_{18} = \frac{w^{2}}{2} D_{p} + w D_{q} , & e_{19} = wz D_{p} + 2z D_{q} , & e_{20} = (wz - 2q) D_{t}, \\ e_{21} = e^{z} \cos(w) D_{x} + e^{z} \sin(w) D_{y}, & e_{22} = e^{z} \sin(w) D_{x} - e^{z} \cos(w) D_{y}, \\ e_{23} = (qw- \frac{z w^{2}}{2})D_{p} + (-wz +2q) D_{q}. \end{array}\end{aligned} \end{equation}$

(5.57)

We implement the following change of basis:

$\begin{equation} \begin{array}{llll} \overline{e}_{1} = e_{2}, & \overline{e}_{2} = e_{6}, & \overline{e}_{3} = e_{8} , & \overline{e}_{4} = e_{9}, \\ \overline{e}_{5} = e_{10}, & \overline{e}_{6} = e_{11}, & \overline{e}_{7} = e_{12}, & \overline{e}_{8} = e_{13}, \\ \overline{e}_{9} = e_{14}, & \overline{e}_{10} = e_{18}, & \overline{e}_{11} = e_{19}, & \overline{e}_{12} = e_{3}, \\ \overline{e}_{13} = e_{5} , & \overline{e}_{14} = e_{21}, & \overline{e}_{15} = e_{22} , & \overline{e}_{16} = e_{1}, \\ \overline{e}_{17} = e_{4}, & \overline{e}_{18} = e_{7} + \frac{e_{23}}{2}, & \overline{e}_{19} = e_{15} , & \overline{e}_{20} = e_{16}, \\ \overline{e}_{21} = e_{7} - \frac{e_{23}}{2}, & \overline{e}_{22} = e_{17}, & \overline{e}_{23} = e_{20}, \end{array} \end{equation}$

(5.58)

and the nonzero brackets of the symmetry algebra are given by

$\begin{equation} \begin{aligned} \begin{array}{lll} \left[e_{2} , e_{9}\right] = e_{1} , & \left[e_{2} , e_{18}\right] = e_{2} + \frac{ e_{6}}{2}, & \left[e_{2}, e_{21}\right] = - e_{2} - \frac{ e_{6}}{2} , \\ \left[e_{2}, e_{23}\right] = -2 e_{3}, & \left[e_{3}, e_{4}\right] = e_{1}, & \left[e_{3}, e_{18}\right] = e_{3}, \\ \left[e_{3}, e_{21}\right] = e_{3}, & \left[e_{3}, e_{22}\right] = 2 e_{2} + e_{6}, & \left[e_{4}, e_{7}\right] = - e_{6}, \\ \left[e_{4}, e_{8}\right] = - e_{5}, & \left[e_{4}, e_{18}\right] = - e_{4}, & \left[e_{4}, e_{21}\right] = - e_{4} , \\ \left[e_{4}, e_{23}\right] = - e_{11} + 2 e_{9} , & \left[e_{5}, e_{16}\right] = - e_{1}, & \left[e_{6}, e_{17}\right] = - e_{1}, \\ \left[e_{7}, e_{17}\right] = - e_{3}, & \left[e_{7}, e_{18}\right] = e_{7}, & \left[e_{7}, e_{21}\right] = e_{7}, \\ \left[e_{7}, e_{22}\right] = 2 e_{10}, & \left[e_{8}, e_{16}\right] = - e_{3}, & \left[e_{8}, e_{18}\right] = e_{8}, \\ \left[e_{8}, e_{21}\right] = e_{8}, & \left[e_{8}, e_{22}\right] = e_{11}, & \left[e_{9}, e_{10}\right] = - e_{6}, \\ \left[e_{9}, e_{11}\right] = -2 e_{5}, & \left[e_{9}, e_{16}\right] = - e_{2}, & \left[e_{9}, e_{18}\right] = e_{11} - e_{9}, \\ \left[e_{9}, e_{21}\right] = - e_{11}+ e_{9}, & \left[e_{9}, e_{22}\right] = -2 e_{4}, & \left[e_{9}, e_{23}\right] = -2 e_{8}, \\ \left[e_{10}, e_{17}\right] = - e_{2} - e_{6} , & \left[e_{10}, e_{18}\right] = e_{10}, & \left[e_{10}, e_{21}\right] = - e_{10}, \\ \left[e_{10}, e_{23}\right] = -2 e_{7}, & \left[e_{11}, e_{16}\right] = -2 e_{2} - e_{6} , & \left[e_{11}, e_{17}\right] = - e_{5}, \\ \left[e_{11}, e_{18}\right] = e_{11}, & \left[e_{11}, e_{21}\right] = - e_{11}, & \left[e_{11}, e_{23}\right] = -4 e_{8}, \\ \left[e_{12}, e_{19}\right] = e_{12}, & \left[e_{12}, e_{20}\right] = - e_{13}, & \left[e_{13}, e_{19}\right] = e_{13}, \\ \left[e_{13}, e_{20}\right] = e_{12}, & \left[e_{14}, e_{16}\right] = - e_{14}, & \left[e_{14}, e_{17}\right] = e_{15}, \\ \left[e_{14}, e_{19}\right] = e_{14}, & \left[e_{14}, e_{20}\right] = e_{15}, & \left[e_{15}, e_{16}\right] = - e_{15}, \\ \left[e_{15}, e_{17}\right] = - e_{14}, & \left[e_{15}, e_{19}\right] = e_{15}, & \left[e_{15}, e_{20}\right] = - e_{14}, \\ \left[e_{16}, e_{18}\right] = -\frac{ e_{10}}{2}, & \left[e_{16}, e_{21}\right] = \frac{ e_{10}}{2}, & \left[e_{16}, e_{23}\right] = e_{7}, \\ \left[e_{17}, e_{18}\right] = - \frac{ e_{11}}{2}+\frac{ e_{9}}{2}, & \left[e_{17}, e_{21}\right] = \frac{ e_{11}}{2}-\frac{ e_{9}}{2}, & \left[e_{17}, e_{22}\right] = e_{4}, \\ \left[e_{17}, e_{23}\right] = e_{8}, & \left[e_{21}, e_{22}\right] = 2 e_{22}, & \left[e_{21}, e_{23}\right] = -2 e_{23}, \\ \left[e_{22}, e_{23}\right] = -4 e_{21}. \end{array}\end{aligned} \end{equation}$

(5.59)

We describe the symmetry algebra by the following proposition:

Proposition 13. The symmetry Lie algebra is a twenty-three-dimensional semi-direct product of twenty-dimensional solvable Lie algebra $S_{2, 20}$ , and $sl(2, \mbox{ $\Bbb R$ })$ . The nilradical is a fifteen-dimensional nilpotant Lie algebra $N_{2, 11} \oplus \mbox{ $\Bbb R$ }^4$ , which is a direct sum of $N_{2, 11}$ , an eleven-dimensional nilpotent Lie algebra, and a four-dimensional abelian Lie algebra $\mbox{ $\Bbb R$ }^4$ . The complement of the nilradical is four-dimensional non-abelian. Therefore, the symmetry Lie algebra can be identified as $S_{2, 20}\rtimes sl(2, \mbox{ $\Bbb R$ })$ .

6. Conclusions and future work

In this work, we have investigated the symmetry Lie algebra of the geodesic equations of the canonical connection on a Lie group corresponding to the eight classes of Lie algebra $A_{6, 20}$ – $A_{6, 27}$ in ^[7]. In each case, we list the nonzero brackets of the given Lie algebra, the geodesic equations, and a basis for the symmetry Lie algebra in terms of vector fields. For every symmetry Lie algebra, we identify its nilradical, solvable complement, and semi-simple factor; a summary of our results is given in . In future work, we plan to study the symmetry Lie algebras for the rest of the six-dimensional Lie algebras $A_{6, 28}$ – $A_{6, 40}$ in ^[7]. The results help to put symmetry Lie algebras into context since they are of very high dimension. It remains to use the symmetries to help integrate the geodesic equations. Another useful by-product is the construction of many large dimensional Levi decomposition Lie algebras, which is a topic of independent interest.

Table 1. Six-dimensional Lie algebras and identification of the symmetry algebra.

Six-dimensional Lie algebras	Dimension	Identification
$A^{ab}_{6, 20}\ (ab: a^{2}+b^{2} \not = 0)$	21	$(\mbox{ $\Bbb R$ }^{12} \rtimes \mbox{ $\Bbb R$ }^{6}) \rtimes sl(2, \mbox{ $\Bbb R$ })$
$A^{a}_{6, 21}$	21	$((A_{5, 1}\oplus \mbox{ $\Bbb R$ }^8) \rtimes \mbox{ $\Bbb R$ }^5) \rtimes sl(2, \mbox{ $\Bbb R$ })$
$A^{\epsilon=0}_{6, 22}$	26	$(\mbox{ $\Bbb R$ }^{13} \rtimes \mbox{ $\Bbb R$ }^5) \rtimes sl(3, \mbox{ $\Bbb R$ })$
$A^{\epsilon=1}_{6, 22}$	23	$(R^{15} \rtimes \mbox{ $\Bbb R$ }^5) \rtimes sl(2, \mbox{ $\Bbb R$ })$
$A^{a, \epsilon = 0}_{6, 23}$	26	$(\mbox{ $\Bbb R$ }^{13} \rtimes \mbox{ $\Bbb R$ }^5) \rtimes sl(3, \mbox{ $\Bbb R$ })$
$A^{a, \epsilon = 1}_{6, 23}$	23	$S_{1, 20}\rtimes sl(2, \mbox{ $\Bbb R$ })$
$A_{6, 24}$	21	$((A_{5.1} \oplus \mbox{ $\Bbb R$ }^8) \rtimes \mbox{ $\Bbb R$ }^5) \rtimes sl(2, \mbox{ $\Bbb R$ })$
$A^{ab}_{6, 25}\ (ab: a^{2}+b^{2} \not = 0)$	18	$((A_{5, 1}\oplus \mbox{ $\Bbb R$ }^4) \rtimes \mbox{ $\Bbb R$ }^5) \rtimes sl(2, \mbox{ $\Bbb R$ })$
$A^{a}_{6, 26}$	21	$(\mbox{ $\Bbb R$ }^{12} \rtimes \mbox{ $\Bbb R$ }^{6}) \rtimes sl(2, \mbox{ $\Bbb R$ })$
$A^{a=0}_{6, 26}$	23	$(\mbox{ $\Bbb R$ }^{12} \rtimes \mbox{ $\Bbb R$ }^{5}) \rtimes (sl(2, \mbox{ $\Bbb R$ }) \oplus sl(2, \mbox{ $\Bbb R$ }))$
$A^{\epsilon=0}_{6, 27}$	26	$(\mbox{ $\Bbb R$ }^{13} \rtimes \mbox{ $\Bbb R$ }^5) \rtimes sl(3, \mbox{ $\Bbb R$ })$
$A^{\epsilon=1}_{6, 27}$	23	$S_{2, 20}\rtimes sl(2, \mbox{ $\Bbb R$ })$

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Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

Nouf Almutiben would like to thank Jouf University and Virginia Commonwealth University for their support. Ryad Ghanam and Edward Boone would like to thank Qatar Foundation and Virginia Commonwealth University in Qatar for their support through the Mathematical Data Science Lab.

Conflict of interest

The authors declare that they have no conflicts of interest.

References

[1]	B. Căruntu, C. Bota, Approximate polynomial solutions of the nonlinear Lane-Emden type equations arising in astrophysics using the squared remainder minimization method, Comput. Phys. Commun., 184 (2013), 1643–1648. https://doi.org/10.1016/j.cpc.2013.01.023 doi: 10.1016/j.cpc.2013.01.023
[2]	Y. Öztürk, M. Gülsu, An operational matrix method for solving Lane-Emden equations arising in astrophysics, Math. Methods Appl. Sci., 37 (2013), 2227–2235. https://doi.org/10.1002/mma.2969 doi: 10.1002/mma.2969
[3]	A. M. Wazwaz, R. Rach, J. S. Duan, A study on the systems of the Volterra integral forms of the Lane-Emden equations by the Adomian decomposition method, Math. Methods Appl. Sci., 37 (2013), 10–19. https://doi.org/10.1002/mma.2969 doi: 10.1002/mma.2969
[4]	Y. Öztürk, An efficient numerical algorithm for solving system of Lane-Emden type equations arising in engineering, Nonlinear Eng., 8 (2019), 429–437. https://doi.org/10.1515/nleng-2018-0062 doi: 10.1515/nleng-2018-0062
[5]	S. Kumbinarasaiah, G. Manohara, G. Hariharan, Bernoulli wavelets functional matrix technique for a system of nonlinear singular Lane Emden equations, Math Comput Simul., 204 (2023), 133–165. https://doi.org/10.1016/j.matcom.2022.07.024 doi: 10.1016/j.matcom.2022.07.024
[6]	AK Verma, N. Kumar, D. Tiwari, Haar wavelets collocation method for a system of nonlinear singular differential equations, Eng. Comput., 38 (2021), 659–698. https://doi.org/10.1108/EC-04-2020-0181 doi: 10.1108/EC-04-2020-0181
[7]	R. Saadeh, A. Burqan, A. El-Ajou, Reliable solutions to fractional Lane-Emden equations via Laplace transform and residual error function, Alex. Eng. J., 61 (2022), 10551–10562. https://doi.org/10.1016/j.aej.2022.04.004 doi: 10.1016/j.aej.2022.04.004
[8]	O. Ala'yed, R. Saadeh, A. Qazza, Numerical solution for the system of Lane-Emden type equations using cubic B-spline method arising in engineering, AIMS Math., 8 (2023), 14747–14766. https://doi.org/10.3934/math.2023754 doi: 10.3934/math.2023754
[9]	P. Roul, K. Thula, A fourth-order B-spline collocation method and its error analysis for Bratu-type and Lane-Emden problems, Int. J. Comput. Math., 96 (2019), 85–104. https://doi.org/10.1080/00207160.2017.1417592 doi: 10.1080/00207160.2017.1417592
[10]	P. Roul, K. Thula, R. Agarwal, Non-optimal fourth-order and optimal sixth-order B-spline collocation methods for Lane-Emden boundary value problems, Appl. Numer. Math., 145 (2019), 342–360.
[11]	E. Salah, R. Saadeh, A. Qazza, R. Hatamleh, Direct power series approach for solving nonlinear initial value problems, Axioms, 12 (2023), 111. https://doi.org/10.3390/axioms12020111 doi: 10.3390/axioms12020111
[12]	Y. Öztürk, Solution for the system of Lane-Emden type equations using Chebyshev polynomials, Mathematics, 6 (2018), 181. https://doi.org/10.3390/math6100181 doi: 10.3390/math6100181
[13]	M. Izadi, A discontinuous finite element approximation to singular Lane-Emden type equations, Appl. Math. Comput., 401 (2021), 126115. https://doi.org/10.1016/j.amc.2021.126115 doi: 10.1016/j.amc.2021.126115
[14]	J. Shahni, R. Singh, Numerical solution of system of Emden-Fowler type equations by Bernstein collocation method, J Math Chem., 59 (2021), 1117–1138. https://doi.org/10.1007/s10910-021-01235-5 doi: 10.1007/s10910-021-01235-5
[15]	M. Izadi, H. M. Srivastava, An efficient approximation technique applied to a non-linear Lane-Emden pantograph delay differential model, Appl. Math. Comput., 401 (2021), 126123. https://doi.org/10.1016/j.amc.2021.126123 doi: 10.1016/j.amc.2021.126123
[16]	M. Abdelhakem, M. Fawzy, M. El-Kady, H. Moussa, Legendre polynomials' second derivative tau method for solving Lane-Emden and Ricatti equations, Appl. Math. Inf. Sci., 17 (2023), 437–445. https://doi.org//10.18576/amis/170305
[17]	M. Abdelhakem, M. Fawzy, M. El-Kady, H. Moussa, An efficient technique for approximated BVPs via the second derivative Legendre polynomials pseudo-Galerkin method: certain types of applications, Results Phys., 43 (2022), 106067. https://doi.org/10.1016/j.rinp.2022.106067 doi: 10.1016/j.rinp.2022.106067
[18]	M. Abdelhakem, H. Moussa, Pseudo-spectral matrices as a numerical tool for dealing BVPs, based on Legendre polynomials' derivatives, Alex. Eng. J., 66 (2023), 301–313. https://doi.org/10.1016/j.aej.2022.11.006 doi: 10.1016/j.aej.2022.11.006
[19]	M. Abdelhakem, Shifted Legendre fractional pseudo-spectral integration matrices for solving fractional Volterra integro-differential equations and Abel's integral equations, Fractals, 2023 (2023), 2340190. https://doi.org/10.1142/S0218348X23401904 doi: 10.1142/S0218348X23401904
[20]	D. Abdelhamid, W. Albalawi, K. S. Nisar, A. Abdel-Aty, S. Alsaeed, M. Abdelhakem, Mixed Chebyshev and Legendre polynomials differentiation matrices for solving initial-boundary value problems, AIMS Math., 8 (2023), 24609–24631. https://doi.org/10.3934/math.20231255 doi: 10.3934/math.20231255
[21]	D. Abdelhamied, M. Abdelhakem, M. El-Kady, Y. H. Youssri, Adapted shifted Chebyshev operational matrix of derivatives: two algorithms for solving even-order BVPs, Appl. Math. Inf. Sci., 17 (2023), 505–511. https://doi.org/10.18576/amis/170318 doi: 10.18576/amis/170318
[22]	O. Ala'yed, B. Batiha, D. Alghazo, F. Ghanim, Cubic B-spline method for the solution of the quadratic Riccati differential equation, AIMS Math., 8 (2023), 9576–9584. https://doi.org 10.3934/math.2023483
[23]	R. Abdelrahim, Z. Omar, O. Ala'yed, B. Batiha, Hybrid third derivative block method for the solution of general second order initial value problems with generalized one step point, Eur. J. Pure Appl. Math., 12 (2019), 1199–1214. https://doi.org/10.29020/nybg.ejpam.v12i3.3425 doi: 10.29020/nybg.ejpam.v12i3.3425
[24]	O. H. Ala'yed, T. Y. Ying, A. Saaban, New fourth order quartic spline method for solving second order boundary value problems, Matematika, 31 (2015), 149–157. https://doi.org/10.11113/matematika.v31.n2.789 doi: 10.11113/matematika.v31.n2.789
[25]	A. S. Heilat, N. N. Hamid, A. I. M. Ismail, Extended cubic B-spline method for solving a linear system of second-order boundary value problems, SpringerPlus, 5 (2016), 1314. https://doi.org/10.1186/s40064-016-2936-4 doi: 10.1186/s40064-016-2936-4
[26]	O. Ala'yed, T. Y. Ying, A. Saaban, Quintic spline method for solving linear and nonlinear boundary value problems, Sains Malays., 45 (2016), 1007–1012.
[27]	B. Batiha, F. Ghanim, O. Ala'yed, R. E. Hatamleh, A. S. Heilat, H. Zureigat, et al., Solving multispecies Lotka-Volterra equations by the Daftardar-Gejji and Jafari method, Int. J. Math. Math. Sci., 2022 (2022), 1839796. https://doi.org/10.1155/2022/1839796 doi: 10.1155/2022/1839796
[28]	O. Ala'yed, T. Y. Ying, A. Saaban, Numerical solution of first order initial value problem using quartic spline method, AIP Conf. Proc., 1691 (2015), 040003. https://doi.org/10.1063/1.4937053 doi: 10.1063/1.4937053
[29]	M. Al-Towaiq, O. Ala'yed, An efficient algorithm based on the cubic spline for the solution of Bratu-type equation, J. Interdiscip. Math., 17 (2014), 471–484. https://doi.org/10.1080/09720502.2013.842050 doi: 10.1080/09720502.2013.842050
[30]	O. Ala'yed, B. Batiha, R. Abdelrahim, A. A. Jawarneh, On the numerical solution of the nonlinear Bratu type equation via quintic B-spline method, J. Interdiscip. Math., 22 (2019), 405–413. https://doi.org/10.1080/09720502.2019.1624305 doi: 10.1080/09720502.2019.1624305
[31]	A. S. Heilat, B. Batiha, T. Qawasmeh, R. Hatamleh, Hybrid cubic B-spline method for solving a class of singular boundary value problems, Eur. J. Pure Appl. Math., 16 (2023), 751–762. https://doi.org/10.29020/nybg.ejpam.v16i2.4725 doi: 10.29020/nybg.ejpam.v16i2.4725
[32]	S. E. Kutluay, Y. U. Ucar, Numerical solutions of the coupled Burgers' equation by the Galerkin quadratic B‐spline finite element method, Math. Methods Appl. Sci., 36 (2013), 2403–2415. https://doi.org/10.1002/mma.2767 doi: 10.1002/mma.2767
[33]	N. Ezhov, F. Neitzel, S. Petrovic, Spline approximation, part 1: basic methodology, J. Appl. Geod., 12 (2018), 139–55. https://doi.org/10.1515/jag-2017-0029 doi: 10.1515/jag-2017-0029
[34]	N. Ezhov, F. Neitzel, S. Petrovic, Spline approximation, part 2: from polynomials in the monomial basis to B-splines: A derivation, Mathematics, 9 (2021), 2198. https://doi.org/10.3390/math9182198 doi: 10.3390/math9182198
[35]	R. K. Lodhi, H. K. Mishra, Quintic B-spline method for numerical solution of fourth order singular perturbation boundary value problems, Stud. Univ. Babeş-Bolyai Math., 63 (2018), 141–151. https://doi.org/10.24193/subbmath.2018.1.09 doi: 10.24193/subbmath.2018.1.09
[36]	S. Özer, Numerical solution of the Rosenau-KdV-RLW equation by operator splitting techniques based on B‐spline collocation method, Numer. Methods Partial Differ. Equ., 35 (2019), 1928–1943. https://doi.org/10.1002/num.22387 doi: 10.1002/num.22387
[37]	A. M. Nagy, A. A. El-Sayed, A novel operational matrix for the numerical solution of nonlinear Lane-Emden system of fractional order, Comput. Appl. Math., 40 (2021), 85. https://doi.org/10.1007/s40314-021-01477-8 doi: 10.1007/s40314-021-01477-8

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AIMS Mathematics

A quintic B-spline technique for a system of Lane-Emden equations arising in theoretical physical applications

Related Papers:

Abstract

1. Introduction

2. Background concepts

3. The canonical connection on Lie groups

4. Symmetries of the geodesic equations

5. Lie symmetry algebras of $A_{6, 20}$ – $A_{6, 27}$

5.1. Algebra $A^{ab}_{6, 20} (ab$ : $a^{2}+b^{2} \not = 0)$

5.2. Algebra $A^{a}_{6, 21}$

5.3. Algebra $A^{a\epsilon}_{6, 22} (a \epsilon$ : $a^{2}+\epsilon^{2}\not = 0, \epsilon= 0, 1)$

5.3.1. $A^{\epsilon=0}_{6, 22}$

5.3.2. $A^{\epsilon=1}_{6, 22}$

5.4. Algebra $A^{a\epsilon}_{6, 23}(a \epsilon$ : $a \ge 0, \epsilon= 0, 1)$

5.4.1. Case 1: $A^{a, \epsilon = 0}_{6, 23}$

5.5. Case 2: $A^{a, \epsilon = 1}_{6, 23}$

5.5.1. $A^{a \not = 0, \epsilon = 1}_{6, 23}$

5.6. Algebra $A_{6, 24}$

5.7. Algebra $A^{ab}_{6, 25} (ab$ : $a^{2}+b^{2} \not = 0$ )

5.8. Algebra $A^{a}_{6, 26}$

5.8.1. Case 1: $A^{a = 0}_{6, 26}$

5.9. Algebra $A^{\epsilon}_{6, 27}$ : $(\epsilon = 0, 1$ )

5.9.1. $A^{\epsilon=0}_{6, 27}$

5.9.2. $A^{\epsilon=1}_{6, 27}$

6. Conclusions and future work

Use of AI tools declaration

Acknowledgments

Conflict of interest

References

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AIMS Mathematics

A quintic B-spline technique for a system of Lane-Emden equations arising in theoretical physical applications

Related Papers:

Abstract

1. Introduction

2. Background concepts

3. The canonical connection on Lie groups

4. Symmetries of the geodesic equations

5. Lie symmetry algebras of A6,20 A_{6, 20} –A6,27 A_{6, 27}

5.1. Algebra Aab6,20(ab A^{ab}_{6, 20} (ab : a2+b2≠0) a^{2}+b^{2} \not = 0)

5.2. Algebra Aa6,21 A^{a}_{6, 21}

5.3. Algebra Aaϵ6,22(aϵ A^{a\epsilon}_{6, 22} (a \epsilon : a2+ϵ2≠0,ϵ=0,1) a^{2}+\epsilon^{2}\not = 0, \epsilon= 0, 1)

5.3.1. Aϵ=06,22 A^{\epsilon=0}_{6, 22}

5.3.2. Aϵ=16,22 A^{\epsilon=1}_{6, 22}

5.4. Algebra Aaϵ6,23(aϵ A^{a\epsilon}_{6, 23}(a \epsilon : a≥0,ϵ=0,1) a \ge 0, \epsilon= 0, 1)

5.4.1. Case 1: Aa,ϵ=06,23 A^{a, \epsilon = 0}_{6, 23}

5.5. Case 2: Aa,ϵ=16,23 A^{a, \epsilon = 1}_{6, 23}

5.5.1. Aa≠0,ϵ=16,23 A^{a \not = 0, \epsilon = 1}_{6, 23}

5.6. Algebra A6,24 A_{6, 24}

5.7. Algebra Aab6,25(ab A^{ab}_{6, 25} (ab : a2+b2≠0 a^{2}+b^{2} \not = 0 )

5.8. Algebra Aa6,26 A^{a}_{6, 26}

5.8.1. Case 1: Aa=06,26 A^{a = 0}_{6, 26}

5.9. Algebra Aϵ6,27 A^{\epsilon}_{6, 27} : (ϵ=0,1 (\epsilon = 0, 1 )

5.9.1. Aϵ=06,27 A^{\epsilon=0}_{6, 27}

5.9.2. A^{\epsilon=1}_{6, 27} A^{\epsilon=1}_{6, 27}

6. Conclusions and future work

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通讯作者: 陈斌, bchen63@163.com

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5. Lie symmetry algebras of $A_{6, 20}$ – $A_{6, 27}$

5.1. Algebra $A^{ab}_{6, 20} (ab$ : $a^{2}+b^{2} \not = 0)$

5.2. Algebra $A^{a}_{6, 21}$

5.3. Algebra $A^{a\epsilon}_{6, 22} (a \epsilon$ : $a^{2}+\epsilon^{2}\not = 0, \epsilon= 0, 1)$

5.3.1. $A^{\epsilon=0}_{6, 22}$

5.3.2. $A^{\epsilon=1}_{6, 22}$

5.4. Algebra $A^{a\epsilon}_{6, 23}(a \epsilon$ : $a \ge 0, \epsilon= 0, 1)$

5.4.1. Case 1: $A^{a, \epsilon = 0}_{6, 23}$

5.5. Case 2: $A^{a, \epsilon = 1}_{6, 23}$

5.5.1. $A^{a \not = 0, \epsilon = 1}_{6, 23}$

5.6. Algebra $A_{6, 24}$

5.7. Algebra $A^{ab}_{6, 25} (ab$ : $a^{2}+b^{2} \not = 0$ )

5.8. Algebra $A^{a}_{6, 26}$

5.8.1. Case 1: $A^{a = 0}_{6, 26}$

5.9. Algebra $A^{\epsilon}_{6, 27}$ : $(\epsilon = 0, 1$ )

5.9.1. $A^{\epsilon=0}_{6, 27}$

5.9.2. $A^{\epsilon=1}_{6, 27}$