# Article

Ufa Mathematical Journal
Volume 3, Number 4, pp. 113-118

# Nonlinear hyperbolic equations with characteristic ring of dimension $3$

Murtazina R.D.

The paper provides a method of classification of nonlinear hyperbolic equations $u_{xy}=f(u,u_x,u_y)$, Darboux integrable, based on a study of the characteristic pairs of Lie rings. A constructive condition on the right side of the equation $f$ with the characteristic ring of dimension three. These equations have the integrals of the second order. In particular, for the equation $u_{xy}=\varphi(u)\psi(u_1)h(\overline{u}_1)$ is a list that satisfy the conditions of this constructive. Formulas are given $x$ - and $y$-integrals for these equations.