  In the present work, a nonlocal nonlinear Schr\"odinger (NLS) model is
studied by means of a recent technique that identifies solutions of partial
differential equations, by considering them as fixed points in {\it
space-time}. This methodology allows to perform a continuation of well-known
solutions of the local NLS model to the nonlocal case. Four different examples
of this type are presented, namely (a) the rogue wave in the form of the
Peregrine soliton, (b) the generalization thereof in the form of the
Kuznetsov-Ma breather, as well as two spatio-temporally periodic solutions in
the form of elliptic functions. Importantly, all four waveforms can be
continued in intervals of the parameter controlling the nonlocality of the
model. The first two can be continued in a narrower interval, while the
periodic ones can be extended to arbitrary nonlocalities and, in fact, present
an intriguing bifurcation whereby they merge with (only) spatially periodic
structures. The results suggest the generic relevance of rogue waves and
related structures, as well as periodic solutions, in nonlocal NLS models.
