On the classification of nonlinear integrable three-dimensional chains via characteristic Lie algebras

We continue describing integrable nonlinear chains of the form $u^j_{n+1,x}=u^j_{n,x}+f(u^{j+1}_{n},u^{j}_n,u^j_{n+1 },u^{j-1}_{n+1})$ with three independent variables on the basis of the existence of a hierarchy of Darboux-integrable reductions. The classification algorithm is based on the well-known fact that characteristic algebras of Darboux-integrable systems have a finite dimension. We use a characteristic algebra in the $x$-direction, whose structure for a given class of models is defined by some polynomial $P(\lambda)$ of degree not exceeding $3$ in the known examples. We assume that $P(\lambda)=\lambda^2$, the classification problem in that case reduces to finding eight unknown functions of a single variable. We obtain a rather narrow class of candidates for the integrability.