  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 function $f(u_{n+1}, u_n, u_{n-1})$
is a quasi-polynomial satisfying the following equation
$\frac{\partial^2}{\partial u_{n+1}\partial u_{n-1}}f(u_{n+1}, u_n,
u_{n-1})=Ce^{{\alpha}u_n-{\frac{{\alpha}m}{2}}u_{n+1}-{\frac{{\alpha}k}{2}}u_{n-1}},$
where $C$ and $\alpha$ are constant parameters and $k,\,m$ are nonnegative
integers.
