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The extension of the Painlev\'e-Calogero coorespondence for n-particleInozemtsev systems raises to the multi-particle generalisations of thePainlev\'e equations which may be obtained by the procedure of Hamiltonianreduction applied to the matrix or non-commutative Painlev\'e systems, whichalso gives isomonodromic formulation for these non-autonomous Hamiltoniansystems. We provide here dual systems for the rational multi-particlePainlev\'e systems (PI,PII and PIV) by reduction from another intersection acoadjoint orbit of GL(n) action with the level set of moment map. We describethis duality in terms of the spectral curve of non-reduced system in comparisonto the Ruijsenaars duality.

Dualities for rational multi-particle Painlev\'e systems: Spectral versus Ruijsenaars