  In the framework of nonlinear Hamiltonian lattices, we revisit the proof of
Moser-Darboux's Theorem, in order to present a general scheme for its
constructive applicability to Hamiltonian models with non-standard symplectic
structures. We take as a guiding example the Salerno and Ablowitz-Ladik (AL)
models: we justify the form of a well-known change of coordinates which is
adapted to the Gauge symmetry, by showing that it comes out in a natural way
within the general strategy outlined in the proof. Moreover, the full or
truncated Lie-series technique in the extended phase-space is used to transform
the Salerno model, at leading orders in the Darboux coordinates: thus the dNLS
Hamiltonian turns out to be a normal form of the Salerno and AL models; as a
byproduct we also get estimates of the dynamics of these models by means of
dNLS one. We also stress that, once it is cast into the perturbative approach,
the method allows to deal with the cases where the explicit trasformation is
not known, or even worse it is not writable in terms of elementary functions.
