  Q-conditional symmetries (nonclassical symmetries) for a general class of
two-component reaction-diffusion systems with non-constant diffusivities are
studied. The work is a natural continuation of our paper (Cherniha and
Davydovych, 2012) in order to extend the results on so-called no-go case. Using
the notion of Q-conditional symmetries of the first type, an exhaustive list of
reaction-diffusion systems admitting such symmetry is derived. The results
obtained are compared with those derived earlier. The symmetries for reducing
reaction-diffusion systems to two-dimensional dynamical systems (ODE systems)
and finding exact solutions are applied. As result, multiparameter families of
exact solutions in the explicit form for nonlinear reaction-diffusion systems
with an arbitrary power-law diffusivity are constructed and their properties
for possible applicability are established.
