Noncommutative Painlevé Equations and Systems of Calogero Type

All Painlevé equations can be written as a time-dependent Hamiltonian system, and as such they admit a natural generalization to the case of several particles with an interaction of Calogero type (rational, trigonometric or elliptic). Recently, these systems of interacting particles have been proved to be relevant in the study of β -models. An almost two-decade-old open question by Takasaki asks whether these multi-particle systems can be understood as isomonodromic equations, thus extending the Painlevé correspondence. In this paper we answer in the affirmative by displaying explicitly suitable isomonodromic Lax pair formulations. As an application of the isomonodromic representation, we provide a construction based on discrete Schlesinger transforms, to produce solutions for these systems for special values of the coupling constants, starting from uncoupled ones; the method is illustrated for the case of the second Painlevé equation.