Painlevé–Calogero correspondence revisited

We extend the work of Fuchs, Painlevé and Manin on a Calogero-like expression of the sixth Painlevé equation (the “Painlevé–Calogero correspondence”) to the other five Painlevé equations. The Calogero side of the sixth Painlevé equation is known to be a nonautonomous version of the (rank one) elliptic model of Inozemtsev’s extended Calogero systems. The fifth and fourth Painlevé equations correspond to the hyperbolic and rational models in Inozemtsev’s classification. Those corresponding to the third, second and first are not included therein. We further extend the correspondence to the higher rank models, and obtain a “multi-component” version of the Painlevé equations.