Nonlinear hyperbolic equations with characteristic ring of dimension 3

The paper provides a method of classification of Darboux integrable nonlinear hyperbolic equations $u_{xy}=f(u,u_x,u_y)$ based on investigation of the characteristic pairs of Lie rings. Constructive conditions on the right-hand side $f$ of the equation with the characteristic ring of the dimension three are obtained. These equations possess second-order integrals. In particular, a list of equations satisfying the constructive conditions is given for the equation $u_{xy}=\varphi(u)\psi(u_x)h(u_y)$. Formulae of $x$- and $y$-integrals are given for these equations.