Ordinary differential equations described by their Lie symmetry algebra

TL;DR

This paper reviews the theory of Lie remarkable equations, focusing on how Lie symmetries characterize certain ordinary differential equations and identifying equations admitted by specific Lie algebras.

Contribution

It applies the theory of Lie remarkable equations to classify ODEs based on their Lie symmetry algebras, providing new characterizations.

Findings

Characterization of Lie remarkable equations via Lie symmetries

Identification of ODEs admitted by specific Lie algebras

Enhanced understanding of symmetry-based classification of differential equations

Abstract

The theory of Lie remarkable equations, i.e. differential equations characterized by their Lie point symmetries, is reviewed and applied to ordinary differential equations. In particular, we consider some relevant Lie algebras of vector fields on $\mathbb{R}^k$ and characterize Lie remarkable equations admitted by the considered Lie algebras.