  A detailed analysis of the small-amplitude solutions of a deformed discrete
nonlinear Schr\"{o}dinger equation is performed. For generic deformations the
system possesses "singular" points which split the infinite chain in a number
of independent segments. We show that small-amplitude dark solitons in the
vicinity of the singular points are described by the Toda-lattice equation
while away from the singular points are described by the Korteweg-de Vries
equation. Depending on the value of the deformation parameter and of the
background level several kinds of solutions are possible. In particular we
delimit the regions in the parameter space in which dark solitons are stable in
contrast with regions in which bright pulses on nonzero background are
possible. On the boundaries of these regions we find that shock waves and
rapidly spreading solutions may exist.
