Painlevé equations and related isomonodromic deformations of linear systems

M. V. Babich<sup>ab

a</sup> St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences
^b Saint Petersburg State University

Abstract: 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.

Keywords: Painlevé equations, isomonodromic deformations, Fuchsian linear systems, non-Fuchsian linear systems, Hamiltonian reduction, confluence of singularities.

MSC: 34M56

Received: 25.01.2025
Revised: 15.03.2025

DOI: 10.4213/im9703

 English version: Izvestiya: Mathematics, 2026, 90:1, 3–33