Integrals and characteristic algebras for systems of discrete equations on a quadrilateral graph

We study the system of discrete equations on the quadrilateral graph. We introduce the notion of the set of independent minimal-order integrals along the characteristic directions, as well as the concept of the characteristic Lie–Rinehart algebra for the system of equations on the graph. We prove that the system admits the complete set of integrals along the considered direction if and only if the dimension of the characteristic algebra corresponding to this direction is finite. In other words, the system is Darboux-integrable if and only if its characteristic algebras in both directions are finite dimensional. As examples of Darboux-integrable systems of discrete equations on quadrilateral graphs we consider reductions of Hirota–Miwa equation, the $Y$-system, and the Kadomtsev–Petviashvili lattice equation and construct the characteristic algebras for them.