Conditional Lie-Bäcklund symmetry and reduction of evolution equations

We suggest a generalization of the notion of invariance of a given partial differential equation with respect to Lie-Backlund vector field. Such generalization proves to be effective and enables us to construct principally new Ansatze 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 can not be constructed by means of the symmetry reduction procedure