Exact radially-symmetric solutions of a class of nonlinear elliptic systems of equations

We study the system of two equations of elliptic type with two nonlinearities depending on the sum of squares of sought-for functions. We obtain conditions on nonlinearities with which the system is reduced to one equation. We also find parametric families of exact solutions, both radially symmetric, and anisotropic with respect to spatial variables, described by elementary or harmonic functions. In case of controlled nonlinearity we specify wide class of realizable exact solutions expressed via harmonic functions.