A note on the Lax pairs for Painleve equations

For the classical Painleve equations, besides the method of similarity redu ction of Lax pairs for integrable partial differential equations, two ways are known for Lax pair generation. The first is based on the confluence pro cedure in Fuchs' linear ODE with four regular singularities isomonodromy de formation which is governed by the sixth Painleve equation. The second meth od treats the hypergeometric equation and confluent hypergeometric equation s as the isomonodromy deformation equations for the triangular systems of O DEs, in whose non-triangular extensions give rise to the Lax pairs for the Painleve equations. The theory of integrable integral operators suggests a new way of Lax pair generation for the classical Painleve equations. This method involves a spe cial kind of gauge transformation that is applied to linear systems which a re exactly solvable in terms of the classical special functions. Some of th e Lax pairs we introduce are known, others are new. The question of gauge e quivalence of different Lax pairs for the Painleve equations is considered as well.