Lax pairs for the Ablowitz-Ladik system via orthogonal polynomials on the unit circle

Nenciu and Simon found that the analogue of the Toda system in the context of orthogonal polynomials on the unit circle is the defocusing Ablowitz-Ladik system. In this paper we use the CMV and extended CMV matrices defined by Cantero, Moral, and Velazquez, and by Simon, respectively, to construct Lax pair representations for this system in the periodic, finite, and infinite settings.