  In the present work we revisit the Salerno model as a prototypical system
that interpolates between a well-known integrable system (the Ablowitz-Ladik
lattice) and an experimentally tractable non-integrable one (the discrete
nonlinear Schr\"odinger model). The question we ask is: for "generic" initial
data, how close are the integrable to the non-integrable models? Our more
precise formulation of this question is: how well is the constancy of formerly
conserved quantities preserved in the non-integrable case? Upon examining this,
we find that even slight deviations from integrability can be sensitively felt
by measuring these formerly conserved quantities in the case of the Salerno
model. However, given that the knowledge of these quantities requires a deep
physical and mathematical analysis of the system, we seek a more "generic"
diagnostic towards a manifestation of integrability breaking. We argue, based
on our Salerno model computations, that the full spectrum of Lyapunov exponents
could be a sensitive diagnostic to that effect.
