  We derive stationary solutions to the two-dimensional hyperbolic discrete
nonlinear Schr\"odinger (HDNLS) equation by starting from the anti-continuum
limit and extending solutions to include nearest-neighbor interactions in the
coupling parameter. We use pseudo-arclength continuation to capture the
relevant branches of solutions and explore their corresponding stability and
dynamical properties (i.e., their fate when unstable). We focus on nine primary
types of solutions: single site, double site in- and out-of-phase, squares with
four sites in-phase and out-of phase in each of the vertical and horizontal
directions, four sites out-of-phase arranged in a line horizontally, and two
additional solutions having respectively six and eight nonzero sites. The
chosen configurations are found to merge into four distinct bifurcation events.
We unveil the nature of the bifurcation phenomena and identify the critical
points associated with these states and also explore the consequences of the
termination of the branches on the dynamical phenomenology of the model.
