The concept of point conditional (nonclassical) symmetry, which generalizes classical symmetry, is presented. Solutions to nonlinear wave and diffusion equations are obtained using the conditional symmetry reduction method. These solutions cannot be derived within the framework of the classical Lie approach. Using the nonlinear wave equation as an example, it is shown that conditional symmetry operators can be employed to construct Lie–Bäcklund transformations.

We also consider generalized symmetries and reductions of nonlinear evolution equations, as well as generalized separation of variables for the equations under study. The inverse symmetry reduction method is introduced, and its application to systems of evolution equations and the four-dimensional Plebanski (second heavenly) equation is demonstrated.

Additionally, we study the linearization of fully nonlinear second-order partial differential equations via the symmetry method, as well as the integrability in quadratures of second-order ordinary differential equations whose coefficients are expressed in terms of stationary solutions of the Calogero–Bogoyavlenskii–Schiff equation.