  It is shown that both the sinh--Gordon equation and the elliptic Tzitzeica
equation can be interpreted as the Taubes equation for Abelian vortices on a
CMC surface embedded in $\R^{2, 1}$, or on a surface conformally related to a
hyperbolic affine sphere in $\R^3$. In both cases the Higgs field and the U(1)
vortex connection are constructed directly from the Riemannian data of the
surface corresponding to the sinh--Gordon or the Tzitzeica equation. Radially
symmetric solutions lead to vortices with a topological charge equal to one,
and the connection formulae for the resulting third Painlev\'e transcendents
are used to compute explicit values for the strength of the vortices.
