  The Salerno model is a discrete variant of the celebrated nonlinear
Schr\"odinger (NLS) equation interpolating between the discrete NLS (DNLS)
equation and completely integrable Ablowitz-Ladik (AL) model by appropriately
tuning the relevant homotopy parameter. Although the AL model possesses an
explicit time-periodic solution known as the Kuznetsov-Ma (KM) breather, the
existence of time-periodic solutions away from the integrable limit has not
been studied as of yet. It is thus the purpose of this work to shed light on
the existence and stability of time-periodic solutions of the Salerno model. In
particular, we vary the homotopy parameter of the model by employing a
pseudo-arclength continuation algorithm where time-periodic solutions are
identified via fixed-point iterations. We show that the solutions transform
into time-periodic patterns featuring small, yet non-decaying far-field
oscillations. Remarkably, our numerical results support the existence of
previously unknown time-periodic solutions {\it even} at the integrable case
whose stability is explored by using Floquet theory. A continuation of these
patterns towards the DNLS limit is also discussed.
