Deforming Lie algebras to Frobenius integrable non-autonomous Hamiltonian systems

TL;DR

This paper investigates how finite-dimensional Lie algebras of Hamiltonian vector fields can be deformed into Frobenius integrable non-autonomous systems, with applications to Painlevé hierarchies and quasi-Stäckel systems.

Contribution

It provides sufficient conditions for deforming Lie algebras into Frobenius integrable systems, advancing the understanding of non-autonomous Hamiltonian dynamics.

Findings

Established criteria for Lie algebra deformation to Frobenius integrable systems.

Applied results to Painlevé hierarchies and quasi-Stäckel systems.

Enhanced methods for analyzing non-autonomous Hamiltonian systems.

Abstract

Motivated by the theory of Painlev\'e equations and associated hierarchies, we study non-autonomous Hamiltonian systems that are Frobenius integrable. We establish sufficient conditions under which a given finite-dimensional Lie algebra of Hamiltonian vector fields can be deformed to a time-dependent Lie algebra of Frobenius integrable vector fields spanning the same distribution as the original algebra. The results are applied to quasi-St\"ackel systems.