Hermitian and Pseudo-Hermitian Reduction of the GMV Auxiliary System. Spectral Properties of the Recursion Operators

We consider simultaneously two different reductions of a Zakharov-Shabats spectral problem in pole gauge. Using the concept of gauge equivalence, we construct expansions over the eigenfunctions of the recursion operators related to the afore-mentioned spectral problem with arbitrary constant asymptotic values of the potential functions. In doing this, we take into account the discrete spectrum of the scattering operator. Having in mind the applications to the theory of the soliton equations associated to the GMV systems, we show how these expansions modify depending on the symmetries of the functions we expand.