  The article continues the work on the description of integrable nonlinear
chains with three independent variables of the following form
$u^j_{n+1,x}=u^j_{n,x}+f(u^{j+1}_{n}, u^{j}_n,u^j_{n+1 },u^{j-1}_{n+1})$ by the
presence of a hierarchy of reductions integrable in the sense of Darboux,
started in (1). The classification algorithm is based on the well-known fact
that the characteristic algebras of Darboux integrable systems have a finite
dimension. In this paper, we used the characteristic algebra in the direction
$x$, whose structure for a given class of models is determined by some
polynomial $P(\lambda)$, whose degree does not exceed three for known examples.
The article assumes that $P(\lambda)=\lambda^2$, in this case the
classification problem is reduced to finding eight unknown functions of one
variable. In the paper, a rather narrow class of candidates for integrability
is obtained, among which there is a new example of an integrable chain.
