Painleve equations and related isomonodromic deformations of linear systems

Not all isomonodromic deformations are described by Painlevé equations, but each Painlevé equation defines an isomonodromic deformation. Obtaining isomonodromy from Painlevé equations is ideologically transparent and easy to verify. Obtaining Painlevé equations from isomonodromy is difficult and not always possible. In the present paper, all Painlevé equations are derived uniformly, without any restrictions on the parameters. To this aim, a special subclass of isomonodromic deformations (Schlesinger ones) is specialized, and only this subclass is considered. The Fuchsian case (Painlevé-VI equation) is considered in details, the remaining Painlevé equations are obtained from Painlevé-VI by the confluence procedure. Isomonodromy is verified by calculation.