Nowadays, there is much interest in constructing exact analytical solutions of differential equations using Lie symmetry methods. Lie devised the method in the 1880s. These methods were substantially developed utilizing modern mathematical language in the 1960s and 1970s by several different groups of authors such as L.V. Ovsiannikov, G. Bluman, and P. J. Olver, and have since been implemented as a software package for symbolic computation on commonly used platforms such as Mathematica and MAPLE.
In this work, we first develop an algorithmic scheme using the MAPLE platform to perform a Lie symmetry algebra identification and validate it on nonlinear systems of five second-order ordinary differential equations, namely the canonical connection systems of geodesic equations associated with the five-dimensional Lie algebras. In each case, the symmetry generators are determined, and the corresponding nonvanishing Lie brackets are computed. Moreover, the algebraic structure of a Lie algebra of symmetries is identified.
Second, we take a theoretical approach and formulate the conditions for a Lie symmetry for the case of the canonical Lie group connection when the Lie algebra is solvable with a one-codimensional abelian nilradical. Such conditions have a complicated system of PDEs, as it has several equations and coefficients to be determined. However, we push the integration of such PDEs as much as possible and investigate to what extent they clarify the concrete results obtained by MAPLE. A comparison with qualitative analysis is demonstrated
