Title:The Painlevé equivalence problem for a constrained 3D system

Abstract:In this paper we propose a geometric approach to study Painlevé 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 Painlevé property from a geometric perspective. Utilising the Painlevé identification algorithm we also relate this system to the Painlevé 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 Painlevé property without the need to restrict it to the invariant hypersurface.