Systematic construction of nonautonomous Hamiltonian equations of Painlevé-type. II. Isomonodromic Lax representation

Abstract

This is the second article in a suite of articles investigating relations between Stäckel-type systems and Painlevé-type systems. In this paper, we construct isomonodromic Lax representations for Painlevé-type systems found in the previous paper by Frobenius integrable deformations of Stäckel-type systems. We first construct isomonodromic Lax representations for Painlevé-type systems in the so-called magnetic representation and then, using a multitime-dependent canonical transformation, we also construct isomonodromic Lax representations for Painlevé-type systems in the nonmagnetic representation. Thus, we prove that the Frobenius integrable systems constructed in Part I are indeed of Painlevé-type. We also present isomonodromic Lax representations for all one-, two-, and three-dimensional Painlevé-type systems originating in our scheme. Based on these results we propose complete hierarchies of (Formula presented.) that follow from our construction.