On Uniqueness of Submaximally Symmetric Vector Ordinary Differential Equations of C-Class

TL;DR

This paper classifies submaximally symmetric vector ODEs of C-class, extending classical scalar ODE results by analyzing fundamental invariants and harmonic curvature within Cartan geometry.

Contribution

It provides a local classification of submaximally symmetric vector ODEs of C-class with specific invariants, advancing harmonic theory for these equations.

Findings

Classification of submaximally symmetric C-class vector ODEs.

Explicit identification of harmonic 2-cochains for each irreducible C-class module.

Generalization of classical results for scalar ODEs to vector ODEs.

Abstract

The fundamental invariants for vector ODEs of order $\ge 3$ considered up to point transformations consist of generalized Wilczynski invariants and C-class invariants. An ODE of C-class is characterized by the vanishing of the former. For any fixed C-class invariant ${\mathcal U}$, we give a local (point) classification for all submaximally symmetric ODEs of C-class with ${\mathcal U} \not \equiv 0$ and all remaining C-class invariants vanishing identically. Our results yield generalizations of a well-known classical result for scalar ODEs due to Sophus Lie. Fundamental invariants correspond to the harmonic curvature of the associated Cartan geometry. A key new ingredient underlying our classification results is an advance concerning the harmonic theory associated with the structure of vector ODEs of C-class. Namely, for each irreducible C-class module, we provide an explicit identification of a lowest weight vector as a harmonic 2-cochain.