On one integrable discrete system

In this paper, we study a system of nonlinear equations on a square graph related to the affine algebra $A^{(1)}_1$. This system is the simplest representative of the class of discrete systems corresponding to affine Lie algebras. We find the Lax representation and construct hierarchies of higher symmetries. In neighborhoods of singular points $\lambda=0$ and $\lambda=\infty$, we construct formal asymptotic expansions of eigenfunctions of the Lax pair and, based on these expansions, find series of local conservation laws for the system considered.