
Symmetry in Nonlinear Mathematical Physics  2009
Mikhail B. Sheftel (Bogazici University, Istanbul,
Turkey & North Western State Technical University, St. Petersburg, Russia)
On classification of 2ndorder 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 secondorder 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.

