Abstract <p>The paper deals with the diffusion-wave solutions to a system of two degenerate nonlinear parabolic equations. The problem of diffusion wave initiation is considered for arbitrary nonlinearity in the equations of the system and for arbitrary directions of motion of the zero fronts of two unknown functions. For this problem, a theorem on the existence for four different (depending on the directions of motion of the zero fronts) analytical solutions is proved. A new numerical method is proposed which allows the problem to be solved for the first time for the case of oppositely moving zero fronts. A new exact solution is constructed explicitly and used to verify the calculation results. A computational experiment is performed showing the convergence of the numerical method and its effectiveness for various problem parameters.</p>