Conditional Lie-Backlund symmetry and reduction of evolution equations

We suggest a generalization of the notion of invariance of a given partial differential equation with respect to a Lie-Backlund vector field. Such a generalization proves to be effective and enables us to construct principally new ansatz reducing evolution-type equations to several ordinary differential equations. In the framework of the said generalization, we obtain principally new reductions of a number of nonlinear heat conductivity equations u_t=u_xx+F(u,u_x) with poor Lie symmetry and obtain their exact solutions. It is shown that these solutions cannot be constructed by means of the symmetry reduction procedure.