Principal results

• Development of the efficient method for studying conditional symmetries of the nonlinear Dirac, Levi-Leblond and Yang-Mills equations that enables obtaining their new reductions and new exact solutions. Solutions constructed in this way include arbitrary functions and cannot, in principle, be found within the framework of the traditional Lie's approach (see the items A-1,2; B-3,10,14,25,30 of the Publication list).
• Complete solution of the classical problem of investigating compatibility and integrating the (1+3)-dimensional nonlinear d'Alembert-eikonal system (considered for the first time by E.Cartan and H.Bateman). Compatibility criterion of the above system is established and for all cases when it is compatible the general solutions are constructed (see the items A-2; B-7,13,15,32,38; of the Publication list).
• Construction of new conditional symmetries and exact solutions of the multi-dimensional d'Alembert equation in (1+3) dimensions (see the items A-1,2; B-7,15,31,38,46,52 of the Publication list ).
• Establishment of an intimate connection between conditional symmetry of a PDE and its reducibility to lower dimensional PDEs within the framework of the ansatz (direct) approach (see the items A-1,2; B-16,27,36,52,58,60 of the Publication list ).
• Introduction of the new concept of higher-order conditional symmetry that is responcible for a possibility to carry out nonlinear separation of variables and anti-reduction in PDEs. Proof of the theorem about the one-to-one correspondence between reducibility of a one-dimensional evolution PDE to a system of ordinary differential equations and its higher-order conditional symmetry. (see the items B-19,20,27,58,60 of the Publication list ).
• Development of the efficient approach to reduction of initial value problems for nonlinear evolution equations to Cauchy problems for some system of ODEs (the approach is purely algebraic and is mainly based on higher-order conditional symmetry of the evolution equation under study). Classification of initial value problems for a broad class of nonlinear second-order evolution equations that admit dimensional reduction to Cauchy problems (see the items B-58,60 of the Publication list ).
• Proof of the "no-go" theorem in the general theory of conditional symmetries, which states that a problem of finding a conditional symmetry of one-dimensional evolution equation, such that the coefficient of the time derivative in the symmetry operator is equal to zero, is equivalent to solving the evolution equation in question (see the item B-45 of the Publication list ).