  We consider exact and asymptotic solutions of the stationary cubic nonlinear
Schr\"odinger equation (NLSE) on metric graphs. We focus on some basic example
graphs. The asymptotic solutions are obtained using the canonical perturbation
formalism developed in our earlier paper \cite{paper1}. For closed example
graphs (interval, ring, star graph, tadpole graph) we calculate spectral curves
and show how the description of spectra reduces to known characteristic
functions of linear quantum graphs in the low intensity limit. Analogously for
open examples we show how nonlinear scattering of stationary waves arises and
how it reduces to known linear scattering amplitudes at low intensities. In the
short-wave length asymptotics we discuss how genuine nonlinear effects such as
may be described using the leading order of canonical perturbation theory:
bifurcation of spectral curves (and the corresponding solutions) in closed
graphs and multistability in open graphs.
