The inverse symmetry problem for a 2D2D generalized second order evolutionary equation

The paper is devoted to the Lie symmetries of 2D2D nonlinear dynamical systems described by second order partial differential equations. By imposing the invariance condition of the equations under the action of the Lie symmetry operator we obtained a determining system which could be solved in two directions: (i) to find the symmetries of a concrete equation; (ii) to obtain all the compatible equations with an imposed form of symmetry algebra. This paper will pay attention to the indirect problem (ii) and will use the algorithm for determining the most general 2D2D equations which belong, following their symmetries, to the same class with two interesting and important physical models: the nonlinear heat equation and the transfer equation with power-law nonlinearities.