Generalized symmetries of partial differential equations and quasi-exact solvability

Using the adjoint action of infinitesimal translations (with respect to some (in)dependent variables) on specific finite-dimensional subspaces of the space of generalized symmetries of a system of partial differential equations, we explicitly determine the dependence of coefficients of generalized symmetries from these subspaces on the variables mentioned above. We establish the connection of our results with the theory of quasi-exactly solvable models.