The Klein-Gordon Equation and Differential Substitutions of the Form v=φ(u,u_x,u_y)

We present the complete classification of equations of the form $u_{xy}=f(u, u_x, u_y)$ and the Klein-Gordon equations $v_{xy}=F(v)$ connected with one another by differential substitutions $v=varphi(u,u_x,u_y)$ such that $varphi_{u_x}varphi_{u_y} eq 0$ over the ring of complex-valued variables.