  In our recent papers [1, 2] the algorithm for classification of integrable
equations with three independent variables was proposed. This method is based
on the requirement of the existence of an infinite set of Darboux integrable
reductions and on the notion of the characteristic Lie-Rinehart algebras. The
method was applied for the classification of integrable cases of different
subclasses of equations \[ u_{n,xy} = f(u_{n+1},u_n,u_{n-1}, u_{n,x},u_{n,y}),
\quad -\infty < n < \infty \]
  of special forms [3, 4, 5, 6]. Under this approach the novel integrable chain
\[
  u_{n,xy} = \alpha_n(u_{n,x} - u^2_n - 1)(u_{n,y} - u^2_n - 1) + 2
u_n(u_{n,x}+u_{n,y}-u^2_n - 1) \] was obtained. Here $\alpha_n = \frac{1}{u_n -
u_{n-1}} - \frac{1}{u_{n+1}-u_n}$, the sought function $u_n = u_n(x,y)$ depends
on the real $x,y$ and the integer $n$. In present paper we construct Lax pair
for this chain. To construct the Lax pair, we use the scheme suggested in paper
[7]. We verified that the found Lax pair is not ``fake'' by studying the
periodic reduction of the chain and the Lax pair. For the system of the
hyperbolic type equations obtained from the chain we constructed generalized
symmetry of the second order which has unusual structure. The matter is that
the system admits an integral and the symmetry also depends on this integral.
We made a similar analysis for the Ferapontov-Shabat-Yamilov chain $u_{n,xy} =
\alpha_n u_{n,x}u_{n,y}$ as well.
