A complete description of Q-conditional symmetries of reaction-diffusion-convection equation with arbitrary power nonlinearities is finished. It is shown that the results obtained in the first and second parts of this work (see arXiv: math-ph/0612078 and arXiv: math-ph/0706.0814) cannot be extended on new power nonlinearities arising in the diffusion and convection coefficients.

A wide range of new Q-conditional symmetries for reaction-diffusion systems with power diffusivities is constructed. The relevant non-Lie ansätze to reduce the reactiondiffusion systems to ODE systems and examples of exact solutions are obtained. Relation of the solutions obtained with the development of spatially inhomogeneous structures is discussed. In 1952 A.C. Turing published the remarkable paper [1], in which a revolutionary idea about mechanism of morphogenesis (the development of structures in an organism during the life) has been proposed. Roughly speaking, his idea says that the diffusion process can interact with the chemical reaction process in such a way that this can stimulate the development and growth of different forms and structures in an organism. Moreover this differentiation is often impossible without diffusion, i.e., only one may destabilize the spatially homogeneous structures. This effect is known as the Turing instability and mathematically leads to systems of reaction-diffusion equations with nonlinearities of the special form [2,3]. Since 1952 the reaction-diffusion systems have been extensively studied by means of different mathematical methods, including group-theoretical methods. The main attention was paid to investigation of the two-component RD systems of the formwhere U = U(t, x) and V = V (t, x) are two unknown functions representing the densities of cells and chemicals, F (U, V ) and G(U, V ) are two given functions describing interaction between them and environment, the functions D 1 (U) and D 2 (V ) are the relevant diffusivities (usually they are assumed to be constants or power functions), and the subscripts t and x denote differentiation with respect to these variables. At the present time one can claim that all possible Lie symmetries of (1) with the constant diffusivities are completely described in [4,5,6], while in [7,8,9] it has been done for the non-constant diffusivities.The problem of construction of conditional symmetries for (1) is still not solved even in the case of Q-conditional symmetries (non-classical symmetries). Moreover, to our best knowledge, 1 to appear in J.Phys.A: Math.Theor.

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