Symmetry gaps for higher order ordinary differential equations

TL;DR

This paper determines the submaximal symmetry dimensions for higher order scalar and vector ordinary differential equations using a Cartan-geometric approach, extending known maximal symmetry results.

Contribution

It introduces a method to classify submaximal symmetry dimensions and finer curvature-constrained cases for higher order ODEs.

Findings

Identified submaximal symmetry dimensions for scalar ODEs of order ≥4.

Classified curvature-constrained submaximal symmetry dimensions.

Extended symmetry classification beyond maximal cases.

Abstract

The maximal contact symmetry dimensions for scalar ODEs of order $\ge 4$ and vector ODEs of order $\ge 3$ are well known. Using a Cartan-geometric approach, we determine for these ODEs the next largest realizable (submaximal) symmetry dimension. Moreover, finer curvature-constrained submaximal symmetry dimensions are also classified.