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Painlevé-Calogero correspondence revisited. (English) Zbl 1016.34089
Summary: The author extends the work of Fuchs, Painlevé and Martin 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. The author further extends the correspondence to the higher-rank models, and obtains a “multi-component” version of the Painlevé equations.
MSC:
34M55Painlevé and other special equations; classification, hierarchies
37J15Symmetries, invariants, invariant manifolds, momentum maps, reduction