  Recently, a novel bifurcation technique known as the deflated continuation
method (DCM) was applied to the single-component nonlinear Schr\"odinger (NLS)
equation with a parabolic trap in two spatial dimensions. The bifurcation
analysis carried out by a subset of the present authors shed light on the
configuration space of solutions of this fundamental problem in the physics of
ultracold atoms. In the present work, we take this a step further by applying
the DCM to two coupled NLS equations in order to elucidate the considerably
more complex landscape of solutions of this system. Upon identifying branches
of solutions, we construct the relevant bifurcation diagrams and perform
spectral stability analysis to identify parametric regimes of stability and
instability and to understand the mechanisms by which these branches emerge.
The method reveals a remarkable wealth of solutions: these do not only include
some of the well-known ones including, e.g., from the Cartesian or polar small
amplitude limits of the underlying linear problem but also a significant number
of branches that arise through (typically pitchfork) bifurcations. In addition
to presenting a ``cartography'' of the landscape of solutions, we comment on
the challenging task of identifying {\it all} solutions of such a
high-dimensional, nonlinear problem.
