A note on the Lax pairs for Painlevé equations
Andrei A. Kapaev and Evelyne Hubert
For the classical Painlevé equations, besides the method of similarity reduction of Lax pairs for integrable PDEs, there are known two ways of Lax pair generation. The first of them is based on the confluence procedure in the Fuchs' linear ODE with four regular singularities isomonodromy deformation of which is governed by PVI equation. The second method treats the hypergeometric equation and confluent hypergeometric equations as the isomonodromy deformation equations for the triangular systems of ODEs which nontriangular extensions give rise to the Lax pairs for the Painlevé equations. The theory of integrable integral operators suggests a new way of the Lax pair generation for the classical Painlevé equations. This method involves the gauge transformation of special kind applied to the linear systems exactly solvable in terms of the classical special functions. Some of the Lax pairs we introduce are known, other are new. The question of gauge equivalence of different Lax pairs for the Painlevé equations is considered.