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We consider an auxiliary spectral problem originally introduced by Gerdjikov,Mikhailov and Valchev (GMV system) and its modification called pseudo-Hermitianreduction which is extensively studied here for the first time. We describe theintegrable hierarchies of both systems in a parallel way and constructrecursion operators. Using the concept of gauge equivalence, we constructexpansions over the eigenfunctions of recursion operators. This permits us toobtain the expansions for both GMV systems with arbitrary constant asymptoticvalues of the potential functions in the auxiliary linear problems.

Pseudo-Hermitian Reduction of a Generalized Heisenberg Ferromagnet Equation. I. Auxiliary System and Fundamental Properties