A new approach to deriving Bäcklund transformations

We give a new, surprisingly simple approach to the derivation of Bäcklund transformations. Motivated by the use of integrating factors to solve linear ordinary differential equations, for the nonlinear case this new technique leads to differential relations between equations. Although our interest here is in Painlevé equations, our approach is applicable to nonlinear equations more widely. As a completely new result we obtain a matrix version of a classical mapping between solutions of special cases of the second Painlevé equation. This involves the derivation of a new matrix second Painlevé equation, for which we also present a Lax pair. In addition, we give a matrix version of the Schwarzian second Painlevé equation, again a completely new result. In this way we also discover a new definition of matrix Schwarzian derivative.