We present classification of Q-conditional symmetries for the two-dimensional nonlinear wave equations u tt − u xx = F (t, x, u) and the reductions corresponding to these nonlinear symmetries. Classification of inequivalent reductions is discussed.

We present an approach for construction of functional bases of differential invariants for some infinite-dimensional algebras with coefficients of generating operators depending on arbitrary functions. An example for the infinite-dimensional Poincare-type algebra is given.

It is well-known that symmetry properties are extremely important for choosing differential equations which can be suitable for description of real physical processes.We present functional bases of second-order differential invariants for various representations of some extensions of the Poincaré group and for a set of m scalar functions (e.g., for similarity and conformal groups). These results enable us to describe new classes of nonlinear multi-dimensional invariant equations and to simplify the problem of symmetry classification of second-order scalar partial differential equations.

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