Dualities for rational multi-particle Painlevé systems: Spectral versus Ruijsenaars

Abstract

The extension of the Painlevé–Calogero correspondence for the $n$-particle Inozemtsev systems raises to the multi-particle generalisations of the Painlevé equations. This extension may be obtained by the Hamiltonian reduction applied to the matrix Painlevé systems. Additionally, such procedure gives an isomonodromic formulation for these non-autonomous Hamiltonian systems. We provide dual systems for the rational multi-particle Painlevé systems ($P_{\text{I}}$, $P_{\text{II}}$ and $P_{\text{IV}}$) using the Hamiltonian reduction. We describe this duality in terms of the spectral curve of a non-reduced system compared to the Ruijsenaars duality.