On classification of 2nd-order PDEs possessing partner symmetries

Abstract:
Recently we used partner symmetries in order to obtain noninvariant solutions of the heavenly equations of Plebanski and the corresponding heavenly gravitational metrics with no Killing vectors. Here we give 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 and recursions between partner symmetries. Using point and Legendre transformations, we reduce this general PDE to several simplest canonical forms. Among these, we find the two heavenly equations of Plebanski and two new equations, which we call the mixed heavenly equation and asymmetric heavenly equation. We discover all the point and contact symmetries of the canonical equations to be used in the recursion relations. Finally, on the example of the mixed heavenly equation, we show how partner symmetries produce noninvariant solutions of PDEs by lifting them from invariant solutions.