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