On classification of second-order PDEs possessing partner symmetries

Abstract

Recently we have demonstrated how to use partner symmetries for obtaining noninvariant solutions of the heavenly equations of Plebañski that govern heavenly gravitational metrics. In this paper, we present a classification of scalar second-order PDEs with four variables that possess partner symmetries and contain only second derivatives of the unknown. We present a general form of such a PDE together with recursion relations between partner symmetries. This general PDE is transformed to several simplest canonical forms containing two heavenly equations of Plebañski among them and two other nonlinear equations which we call the mixed heavenly equation and asymmetric heavenly equation. We have calculated all the point and contact symmetries of all the canonical equations which can be used as an input in our recursion relations. On an example of the mixed heavenly equation, we show how to use partner symmetries for obtaining noninvariant solutions of PDEs by a lift from invariant solutions. Finally, we present Ricci-flat self-dual metrics governed by solutions of the mixed heavenly equation and its Legendre transform.