  The main goal of the article is testing a new classification algorithm. To
this end we apply it to a relevant problem of describing the integrable cases
of a subclass of two-dimensional lattices. By imposing the cut-off conditions
$u_{-1}=c_0$ and $u_{N+1}=c_1$ we reduce the lattice
$u_{n,xy}=\alpha(u_{n+1},u_n,u_{n-1})u_{n,x}u_{n,y}$ to a finite system of
hyperbolic type PDE. Assuming that for each natural $N$ the obtained system is
integrable in the sense of Darboux we look for $\alpha$. To detect the Darboux
integrability of the hyperbolic type system we use an algebraic criterion of
Darboux integrability which claims that the characteristic Lie rings of such a
system must be of finite dimension. We prove that up to the point
transformations only one lattice in the studied class passes the test. The
lattice coincides with the earlier found Ferapontov-Shabat-Yamilov equation.
The one-dimensional reduction $x=y$ of this lattice passes also the symmetry
integrability test.
