The Painlev\'e equivalence problem for a constrained 3D system

TL;DR

This paper introduces a geometric method to analyze Painlevé equations from constrained 3D systems, demonstrating their integrability, relation to Painlevé VI, and conjecturing their Painlevé property.

Contribution

It develops a geometric framework for studying constrained 3D systems, relating them to Painlevé VI, and proves integrability of the autonomous limit.

Findings

The system can be restricted to two 2D systems with Painlevé property.

The system is related to Painlevé VI via the identification algorithm.

The autonomous limit is Liouville integrable with elliptic curve level sets.

Abstract

In this paper we propose a geometric approach to study Painlev\'e equations appearing as constrained systems of three first-order ordinary differential equations. We illustrate this approach on a system of three first-order differential equations arising in the theory of semi-classical orthogonal polynomials. We show that it can be restricted to a system of two first-order differential equations in two different ways on an invariant hypersurface. We build the space of initial conditions for each of these restricted systems and verify that they exhibit the Painlev\'e property from a geometric perspective. Utilising the Painlev\'e identification algorithm we also relate this system to the Painlev\'e VI equation and we build its global Hamiltonian structure. Finally, we prove that the autonomous limit of the original system is Liouville integrable, and the level curves of its first integrals are elliptic curves, which leads us to conjecture that the 3D system itself also possesses the Painlev\'e property without the need to restrict it to the invariant hypersurface.