  There are two cases when the nonlinear Schr\"odinger equation (NLSE) with an
external complex potential is well-known to support continuous families of
localized stationary modes: the ${\cal PT}$-symmetric potentials and the Wadati
potentials. Recently Y. Kominis and coauthors [Chaos, Solitons and Fractals,
118, 222-233 (2019)] have suggested that the continuous families can be also
found in complex potentials of the form $W(x)=W_{1}(x)+iCW_{1,x}(x)$, where $C$
is an arbitrary real and $W_1(x)$ is a real-valued and bounded differentiable
function. Here we study in detail nonlinear stationary modes that emerge in
complex potentials of this type (for brevity, we call them W-dW potentials).
First, we assume that the potential is small and employ asymptotic methods to
construct a family of nonlinear modes. Our asymptotic procedure stops at the
terms of the $\varepsilon^2$ order, where small $\varepsilon$ characterizes
amplitude of the potential. We therefore conjecture that no continuous families
of authentic nonlinear modes exist in this case, but "pseudo-modes" that
satisfy the equation up to $\varepsilon^2$-error can indeed be found in W-dW
potentials. Second, we consider the particular case of a W-dW potential well of
finite depth and support our hypothesis with qualitative and numerical
arguments. Third, we simulate the nonlinear dynamics of found pseudo-modes and
observe that, if the amplitude of W-dW potential is small, then the
pseudo-modes are robust and display persistent oscillations around a certain
position predicted by the asymptotic expansion. Finally, we study the authentic
stationary modes which do not form a continuous family, but exist as isolated
points. Numerical simulations reveal dynamical instability of these solutions.

Pattern Formation and Solitons;Optics