Multidimensional exact solutions of a nonlinear system of two parabolic equations

We study a system of two parabolic equations with two nonlinearities depending on the difference of the squares of unknown functions. Some conditions are found for reducibility of a system to a sole equation. The cases are distinguished in which the system can be reduced to a linear heat equation or semilinear equations. We also construct parametric families of exact solutions depending on elementary functions.