On the problem of classifying integrable chains with three independent variables

We discuss a new method for the classification of integrable nonlinear chains with three independent variables using an example of chains in the form u^j_n+1,x=u^j_n,x+f(u^j+1_n,u^j_n,u^j_n+1 ,u^j-1_n+1) . This method is based on reductions having the form of systems of differential–difference Darboux-integrable equations. It is well known that the characteristic algebras of Darboux-integrable systems have a finite dimension. The structure of the characteristic algebra is defined by some polynomial P(λ) . The polynomial degree for the known integrable chains from the class under consideration equals 2 or 3 . A partial classification is performed in the case P(λ)=2 .