  The correspondence between a high-order non symmetric difference operator
with complex coefficients and the evolution of an operator defined by a Lax
pair is established. The solution of the discrete dynamical system is studied,
giving explicit expressions for the resolvent function and, under some
conditions, the representation of the vector of functionals, associated with
the solution for our integrable systems. The method of investigation is based
on the evolutions of the matrical moments.
