Formal diagonalization of a discrete Lax operator and conservation laws and symmetries of dynamical systems

We consider the problem of constructing a formal asymptotic expansion in the spectral parameter for an eigenfunction of a discrete linear operator. We propose a method for constructing an expansion that allows obtaining conservation laws of discrete dynamical systems associated with a given linear operator. As illustrative examples, we consider known nonlinear models such as the discrete potential Korteweg–de Vries equation, the discrete version of the derivative nonlinear Schrödinger equation, the Veselov–Shabat dressing chain, and others. We describe the infinite set of conservation laws for the discrete Toda chain corresponding to the Lie algebra $A_1^{(1)}$. We find new examples of integrable systems of equations on a square lattice.