Doubly periodic waves of a discrete nonlinear Schr\"odinger system with saturable nonlinearity

A system of two discrete nonlinear Schr"odinger equations of the Ablowitz-Ladik type with a saturable nonlinearity is shown to admit a doubly periodic wave, whose long wave limit is also derived. As a by-product, several new solutions of the elliptic type are provided for NLS-type discrete and continuous systems.