Conditional symmetries for systems of PDEs: new definitions and their application for reaction-diffusion systems

New definitions of $Q$-conditional symmetry for systems of PDEs are presented, which generalize the standard notation of non-classical (conditional) symmetry. It is shown that different types of $Q$-conditional symmetry of a system generate a hierarchy of conditional symmetry operators. A class of two-component nonlinear RD systems is examined to demonstrate applicability of the definitions proposed and it is shown when different definitions of $Q$-conditional symmetry lead to the same operators.