On a Certain Class of Hyperbolic Equations with Second-order Integrals

In this paper, we examine a special class of nonlinear hyperbolic equations possessing a second-order $y$-integral. We clarify the structure of $x$-integrals and prove that they are $x$-integrals of a hyperbolic equation with a first-order $y$-integral. We also prove that this class contains the well-known Laine equation.