  Q-conditional symmetries (nonclassical symmetries) for a general class of
two-component reaction-diffusion systems with constant diffusivities are
studied. Using the recently introduced notion of Q-conditional symmetries of
the first type (R. Cherniha J. Phys. A: Math. Theor., 2010. vol. 43., 405207),
an exhaustive list of reaction-diffusion systems admitting such symmetry is
derived. The form-preserving transformations for this class of systems are
constructed and it is shown that this list contains only non-equivalent
systems. The obtained symmetries permit to reduce the reaction-diffusion
systems under study to two-dimensional systems of ordinary differential
equations and to find exact solutions. As a non-trivial example, multiparameter
families of exact solutions are explicitly constructed for two nonlinear
reaction-diffusion systems. A possible interpretation to a biologically
motivated model is presented.
