On a class of hyperbolic equations with third-order integrals

We consider a Goursat problem on classification nonlinear second order hyperbolic equations integrable by the Darboux method. In the work we study a class of hyperbolic equations with second order $y$-integral reduced by an differential substitution to equations with first order $y$-integral. It should be noted that Laine equations are in the considered class of equations. In the work we provide a second order $y$-integral for the second Laine equation and we find a differential substitution relating this equation with one of the Moutard equations.
We consider a class of nonlinear hyperbolic equations possessing first order $y$-integrals and third order $x$-integrals. We obtain three conditions under which the equations in this class possess first order and third order integrals. We find the form of such equations and obtain the formulas for $x$- and $y$-integrals. In the paper we also provide differential substitutions relating Laine equations.