On integrability of Rikkati-type systems of hyperbolic equations

Equations of the type $U_{xy}=U*U_x$ are considered. Here $U(x,y)$ is a $T$-mapped function and $T$ is an algebra over the field $\mathbb C$. It is shown that there are two characteristic Lie algebras $L_x$ and $L_y$ connected with each such equation. A definition of the $\mathbb Z$-graded Lie algebra $\mathfrak G$ corresponding to the equation is given. It is proved that for each of the equations the corresponding algebra $\mathfrak G$ can be taken as a sum of vector spaces $L_x$ and $L_y$ with a commutator between elements of $L_x$ and $L_y$ given by zero-curvature relations. Under assumption that the algebra $T$ has no left ideals, the classifications of the equations with finite dimensional characteristic algebras $L_x$ and $L_y$ is given. All of the equations are Darboux-integrable.