Conservation laws for hyperbolic equations: search algorithm for local preimage with respect to the total derivative

We propose an algorithm, which allows one to eliminate flows from conservation laws for hyperbolic equations by expressing partial derivatives of these flows in terms of the corresponding densities. In particular, the application of this algorithm allows one to prove that the decreasing of order of at least one of Laplace $y$-invariants of the equation $u_{xy}=F(x,y,u,u_x,u_y)$ is a necessary condition for the function $F_{u_y}$ belonged to the image of the total derivative $D_x$ by virtue of this equation. Thus, we obtain constructive necessary conditions for the existence of differential substitutions that transform a hyperbolic equation into a linear equation or into the Klein–Gordon equation.