On a class of second-order PDEs admitting partner symmetries

Recently we have demonstrated how to use partner symmetries for obtaining noninvariant solutions of heavenly equations of Plebañski that govern heavenly gravitational metrics. In this paper, we present a class of scalar second-order PDEs with four variables, that possess partner symmetries and contain only second derivatives of the unknown. We present recursion relations for symmetries for these PDEs. We also present a complete set of simplest canonical forms to which the general PDE with partner symmetries can be transformed by point and Legendre transformations, together with recursions for symmetries of these canonical equations. These canonical forms contain the first and second equations of Plebañski and also new equations which we call mixed heavenly equation and asymmetric heavenly equation. On the 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.