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

The extension of the Painlevé-Calogero coorespondence for n-particle Inozemtsev systems raises to the multi-particle generalisations of the Painlevé equations which may be obtained by the procedure of Hamiltonian reduction applied to the matrix or non-commutative Painlevé systems, which also gives isomonodromic formulation for these non-autonomous Hamiltonian systems. We provide here dual systems for the rational multi-particle Painlevé systems (PI,PII and PIV) by reduction from another intersection a coadjoint orbit of GL(n) action with the level set of moment map. We describe this duality in terms of the spectral curve of non-reduced system in comparison to the Ruijsenaars duality.