Exact solutions to nonlinear equations with quadratic nonlinearity

Based on the concepts of the generalized conditional symmetry and of the in variant subspace, a constructive way to derive nonlinear equations with qua dratic nonlinearity that admit exact solutions as well as many new examples is presented. This type of solution, found by Galaktionov, describes the b lowup phenomena. Examples of exact solutions that do not correspond to the invariant subspace are also presented and explained by the nonlinear genera lized conditional symmetries.