  Radial solutions to the elliptic sinh-Gordon and Tzitzeica equations can be
interpreted as Abelian vortices on certain surfaces of revolution. These
surfaces have a conical excess angle at infinity (in a way which makes them
similar to Elizabethan ruff collars). While they can not be embedded in the
Euclidean 3-space, we will show that they can be globally embedded in the
hyperbolic space. The existence of these hyperbolic embeddings follows from the
asymptotic analysis of a Painleve III ODE.
