Spectral dualities for the rational Painlevé systems

The extension of the Painlevé-Calogero correspondence for n-particle Inozemtsev systems raises to the multi-particle generalisations of the Painlevé equations. This multi-particle Painlevé systems may be obtained by the procedure of Hamiltonian reduction applied to the matrix Painlevé systems. This procedure also gives isomonodromic formulation for the obtained non-autonomous multi-particle Hamiltonian systems. In my talk I will show how to obtain these multi-particle Painlevé systems with isomonodromic problems and introduce the dual systems for the rational Painlevé systems (multi-particle generalisation of PIV, PII and PI). This duality may be described in terms of spectral curve of corresponding isomonodromic problem. I will also show how this duality may be extended for any polynomial matrix Hamiltonian system which allows isospectral or isomonodromic representations. Finally, I will discuss the autonomisation of these multi-particle Painlevé equations which may be viewed as the further degeneration of rational Inozemtsev system and how this degeneration connected with a confluence scheme of the Painlevé systems.