Systematic construction of non-autonomous Hamiltonian equations of Painlev\'{e}-type. II. Isomonodromic Lax representation

TL;DR

This paper constructs isomonodromic Lax representations for Painlevé-type systems derived from Stäckel-type systems, confirming their Painlevé nature and proposing hierarchies of classical Painlevé equations.

Contribution

It introduces explicit isomonodromic Lax representations for Painlevé-type systems in magnetic and non-magnetic forms, establishing their integrability and extending to hierarchies of classical Painlevé equations.

Findings

Constructed Lax representations for Painlevé-type systems.

Proved systems are of Painlevé type via Frobenius integrability.

Proposed hierarchies of P_I to P_IV equations.

Abstract

This is the second article in a suite of articles investigating relations between St\"{a}ckel-type systems and Painlev\'{e}-type systems. In this article we construct isomonodromic Lax representations for Painlev\'{e}-type systems found in the previous paper by Frobenius integrable deformations of St\"{a}ckel-type systems. We first construct isomonodromic Lax representations for Painlev\'{e}-type systems in the so called magnetic representation and then, using a multitime-dependent canonical transformation, we also construct isomonodromic Lax representations for Painlev\'{e}-type systems in the non-magnetic representation. Thus, we prove that the Frobenius integrable systems constructed in Part I are indeed of Painlev\'{e}-type. We also present isomonodromic Lax representations for all one-, two- and three-dimensional Painlev\'{e}-type systems originating in our scheme. Based on these results we propose complete hierarchies of $P_{I}-P_{IV}$ that follow from our construction.