  We propose a modified version of the Ginzburg-Landau energy functional
admitting static solitons and determine all the Painlev\'e-integrable cases of
its Bogomolny equations of a given class of models. Explicit solutions are
determined in terms of the third Painlev\'e transcendents, allowing us to
calculate physical quantities such as the vortex number and the vortex
strength. These solutions can be interpreted as the usual Abelian-Higgs
vortices on surfaces of non-constant curvature with conical singularity.
