Integrability conditions for two-dimensional Toda-like equations

In the article some algebraic properties of nonlinear two-dimensional lattices of the form u(n,xy) = f(u(n+1),u(n),u(n-1)) are studied. The problem of exhaustive description of the integrable cases of this kind lattices remains open. By using the approach, developed and tested in our previous works we adopted the method of characteristic Lie-Rinehart algebras to this case. In the article we derived an effective integrability conditions for the lattice and proved that in the integrable case the functionf(u(n+1),u(n),u(n-1)) is a quasi-polynomial satisfying the following equation partial derivative(2)/partial derivative u(n+1)partial derivative u(n-1) f(u(n+1),u(n),u(n-1))=Ce alpha un-alpha m2un+1-alpha k2un-1, where C and alpha are constant parameters and k, m are nonnegative integers.