Algebraic properties of first integrals for systems of second-order ordinary differential equations of maximal symmetry

Abstract

Symmetries of the first integrals for scalar linear or linearizable secondorder ordinary differential equations (ODEs) have already been derived and shown to exhibit interesting properties. One of these is that the symmetry algebra sl(3, IR) is generated by the three triplets of symmetries of the functionally independent first integrals and its quotient. In this paper, we first investigate the Lie-like operators of the basic first integrals for the linearizable maximally symmetric system of two second-order ODEs represented by the free particle system, obtainable from a complex scalar free particle equation, by splitting the corresponding complex basic first integrals and its quotient as well as their associated symmetries. It is proved that the 14 Lie-like operators corresponding to the complex split of the symmetries of the functionally independent first integrals I1, I2 and their quotient I2/I1 are precisely the Lie-like operators corresponding to the complex split of the symmetries of the scalar free particle equation in the complex domain. Then, it is shown that there are distinguished four symmetries of each of the four basic integrals and their quotients of the two-dimensional free particle system which constitute four-dimensional Lie algebras which are isomorphic to each other and generate the full symmetry algebra sl(4, IR) of the free particle system. It is further shown that the (n + 2)-dimensional algebras of the n + 2 first integrals of the system of n free particle equations are isomorphic to each other and generate the full symmetry algebra sl(n + 2, IR) of the free particle system.