  Recently it was demonstrated that the concept of a spectral singularity (SS)
can be generalized to waves propagating in nonlinear media, like matter waves
or electromagnetic waves in Kerr media. The corresponding solutions represent
nonlinear currents sustained by a localized linear complex potential in a
nonlinear Schr\"odinger equation. A key feature allowing the nonlinear
generalization of a SS is a possibility to reduce a nonlinear current to the
linear limit, where a SS has the unambiguous definition. In the meantime, known
examples of nonlinear modes bifurcating from linear spectral singularities are
few and belong to the specific class of constant-amplitude waves. Here we
propose to extend the class of nonlinear SSs by incorporating solutions whose
amplitudes are inhomogeneous. We show that the continuation from the linear
limit requires a deformation of the complex potential, and this deformation is
not unique. Examples include the deformation preserving the gain-and-loss
distribution and the deformation preserving geometry of the potential. For the
case example of a rectangular potential, we demonstrate that the nonlinear
currents can be divided into two types: solutions of the first type bifurcate
from the linear spectral singularities, and solutions of the second type cannot
be reduced to the linear limit.
