Algebraic structure and maximal dimension of the symmetry algebra for arbitrary systems of ODEs

TL;DR

This paper characterizes the algebraic structure and maximum dimension of symmetry algebras for arbitrary systems of ODEs, extending known results for scalar and low-order systems to higher orders.

Contribution

It provides a complete description of the symmetry algebra structure for systems of ODEs reducible to trivial form, including variational and divergence symmetries, for any order n ≥ 3.

Findings

Determines the structure of symmetry algebras for reducible systems of ODEs.

Establishes the maximal dimension of symmetry algebras for linear and nonlinear ODE systems.

Extends known results to arbitrary order systems of ODEs.

Abstract

It is known for scalar ordinary differential equations, and for systems of ordinary differential equations of order not higher than the third, that their Lie point symmetry algebras is of maximal dimension if and only if they can be reduced by a point transformation to the trivial equation $\mathbf{y}^{(n)}$=0. For arbitrary systems of ordinary differential equations of order $n \geq 3$ reducible by point transformations to the trivial equation, we determine the complete structure of their Lie point symmetry algebras as well as that for their variational, and their divergence symmetry algebras. As a corollary, we obtain the maximal dimension of the Lie point symmetry algebra for any system of linear or nonlinear ordinary differential equations.