A quantum-mechanical model of multi-photon lasers is proposed for arbitrary degeneracy of the multi-photon processes of photon emission and absorption governing the interaction between the active medium and the radiation field. Equations of motion are derived for the mean values of the intensive, slowly varying operators essential to the description of the dynamics. Stationary solutions are found and their stability is studied proving the existence of a non-equilibrium first-order phase transition. The critical value of the inversion parameter required for laser action is shown to vary by a jump depending on the degree of degeneracy of the multi-photon processes of emission and absorption considered.