  A new 1-D discrete nonlinear Schr\"{o}dinger (NLS) Hamiltonian is introduced
which includes the integrable Ablowitz-Ladik system as a limit. The symmetry
properties of the system are studied. The relationship between intrinsic
localized states and the soliton of the Ablowitz-Ladik NLS is discussed. It is
pointed out that a staggered localized state can be viewed as a particle of a
{\em negative} effective mass. It is shown that staggered localized states can
exist in the discrete dark NLS. The motion of localized states and
Peierls-Nabarro pinning are studied.
