Compatibility criteria and general solution of the nonlinear dʹAlembert-Eikonal system

We study integrability of a system of nonlinear partial differential equations consisting of the nonlinear dʹAlembert equation □u = F(u) and nonlinear eikonal equation uxμuxμ = G(u) in the complex Minkowski space R(1, 3). A method suggested makes it possible to establish necessary and sufficient compatibility conditions and construct general solution of the dʹAlembert-eikonal system for all cases when it is compatible. The results obtained can be applied, in particular, to construct principally new (non-Lie, non-similarity) solutions of the nonlinear dʹAlembert and Dirac equations.