  The Jacobs-Rebbi equation arises in many contexts where vortical motion in
two-dimensional ideal media is investigated. Alternatively, it can be derived
in the Abelian Higgs field theory. It is considered non-integrable and
numerical solutions have been found, consisting of localised, robust vortices.
We show in this work that the equation is integrable and provide the Lax pair.
The exact solution is obtained in terms of Riemann \emph{theta} functions.
