Quad-equations and auto-Bäcklund transformations of NLS-type systems

Treating an integrable quad-equation along with its two generalized symmetries as a compatible system allows one to construct an auto-Backlund transformation for solutions of the related NLS-type system. A fixed periodic reduction of the quad-equation yields a quasi-periodic reduction of its generalized symmetries that turns them into differential constraints compatible with the NLS-type system.