Reduction and quotient equations for differential equations with symmetries

The method of reduction previously known in the theory of Hamiltonian systems with symmetries is developed in order to obtain exact group-invariant solutions of systems of partial differential equations. This method leads to representations of quotient equations which are very convenient for the systematic analysis of invariant solutions of boundary value problems. In the case of partially invariant solutions, necessary and sufficient conditions of their invariance with respect to subalgebras of symmetry algebras are given. The concept of partial symmetries of differential equations is considered.