  The existence of a local curve of corotating and counter-rotating vortex
pairs was proven by Hmidi and Mateu in via a desingularization of a pair of
point vortices. In this paper, we construct a global continuation of these
local curves. That is, we consider solutions which are more than a mere
perturbation of a trivial solution. Indeed, while the local analysis relies on
the study of the linear equation at the trivial solution, the global analysis
requires on a deeper understanding of topological properties of the nonlinear
problem. For our proof, we adapt the powerful analytic global bifurcation
theorem due to Buffoni and Toland, to allow for the singularity at the
bifurcation point. For both the corotating and the counter-rotating pairs,
along the global curve of solutions either the angular fluid velocity vanishes
or the two patches self-intersect.
