An alternative method of constructing the formal diagonalization for the discrete Lax operators is proposed which can be used to calculate conservation laws and in some cases generalized symmetries for discrete dynamical systems. Discrete

Abstract. Difference-difference systems are suggested corresponding to the Cartan matrices of any simple or affine Lie algebra. In the cases of the algebrasN these systems are proved to be integrable. For the systems corresponding to the algebras A 2 , A (1) 1 , A (2) 2 generalized symmetries are found. For the systems A 2 , B 2 , C 2 , G 2 , D 3 complete sets of independent integrals are found. The Lax representation for the difference-difference systems corresponding to A N , B N , C N , A (1) 1 , D (2) N are presented.

Differential-difference integrable exponential type systems are studied corresponding to the Cartan matrices of semi-simple or affine Lie algebras. For the systems corresponding to the algebras A 2 , B 2 , C 2 , G 2 the complete sets of integrals in both directions are found. For the simple Lie algebras of the classical series A N , B N , C N and affine algebras of series D

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In the work we describe the structure of integrals of systems of discrete equations. We consider an example of discrete Toda chain corresponding to Lie algebra of series 2 .