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 $P(\lambda)$. 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 $\deg P(\lambda)=2$.