  Persistence of stationary and traveling single-humped localized solutions in
the spatial discretizations of the nonlinear Schrodinger (NLS) equation is
addressed. The discrete NLS equation with the most general cubic polynomial
function is considered. Constraints on the nonlinear function are found from
the condition that the second-order difference equation for stationary
solutions can be reduced to the first-order difference map. The discrete NLS
equation with such an exceptional nonlinear function is shown to have a
conserved momentum but admits no standard Hamiltonian structure. It is proved
that the reduction to the first-order difference map gives a sufficient
condition for existence of translationally invariant single-humped stationary
solutions and a necessary condition for existence of single-humped traveling
solutions. Other constraints on the nonlinear function are found from the
condition that the differential advance-delay equation for traveling solutions
admits a reduction to an integrable normal form given by a third-order
differential equation. This reduction also gives a necessary condition for
existence of single-humped traveling solutions. The nonlinear function which
admits both reductions defines a two-parameter family of discrete NLS equations
which generalizes the integrable Ablowitz--Ladik lattice.
