First integrals and exact solutions of a system of ordinary differential equations with power nonlinearity

The system of ordinary differential equations with degree nonlinearities is considered. Systems of such kind arise as comparison systems at stability analysis by means non-linear approximation and at application of reduction method to switched systems. This same kind the equations meet also at construction by reduction method of exact solutions the systems of reaction diffusion modeled by sets of equations in partial derivatives of parabolic type with the degree nonlinearities characterizing reactions of components of mixture. Systems of ordinary differential equations with degree nonlinearities are used in mathematical biology as models of the interacting biological species. We obtain the conditions on parameters of system under which it has explicit exact solutions representable by combination of degree or exponential functions of time. The existence conditions of presented by combinations of degree and logarithmic functions with respect to state variables the first integrals of system are obtained. A number of examples is given, illustrating the received results.