On nonlinear hyperbolic differential equations related to the Klein–Gordon equation by differential substitutions

We present a complete classification of nonlinear hyperbolic differential equations in two independent variables $u_{xy}=f(u,u_x,u_y)$ reduced to the Klein–Gordon equation $v_{xy}=F(v)$ by differential substitutions of the special form $v=\varphi(u,u_x)$.