Systematic construction of non-autonomous Hamiltonian equations of Painlev\'e-type. II. Isomonodromic Lax representation
Abstract
This is the second article in a suite of articles investigating relations between St\"ackel-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\"ackel-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 PI-PIV that follow from our construction.