  Here we present exact, stationary, parametric solutions to the
Schr\"odinger--Poisson system. We confront two images: on one hand, we draw on
the homotopy analysis method which leads us to a nonlinear integral scheme.
Indeed, this approach might be simplified by looking for sufficiently smooth
solutions vanishing asymptotically. However, since our system possesses
stiffness an additional analysis has to be considered. On the other hand, we
seek for exact solutions over the inverse scattering method by introducing a
pseudo spectral transform. In fact, this pseudo spectral method generalises
Korteweg--de Vries family's kernel and let us to circumvent some technical
difficulties originally arisen in our first approach although, again, we come
to an integral representation, which we test for convergence.
