  A method of the formal diagonalization of the discrete linear operator with a
parameter is studied. In the case when the operator provides a Lax operator for
a nonlinear quad system the formal diagonalization method allows one to
describe effectively conservation laws and generalized symmetries for this
system. Asymptotic representation of the Lax operators eigenfunctions are
constructed and infinite series of conservation laws are described for the quad
system connected with $A^{(1)}_3$ affine Lie algebra, for the modified discrete
Boussinesq system and for the discrete Tzitzeica equation. For a newly found
multiquadratic discrete model conservation laws and several generalized
symmetries are presented.
