  We study topological vortex solutions in a generalized Abelian Higgs model
with non-polynomial dielectric and potential functions. These quantities are
chosen by requiring integrability of the self-dual limit of the theory for all
values of the magnetic flux. All the vortex profiles are described by exact
analytical expressions that solve the self-dual vortex equations. There is only
a symmetry-breaking superconducting phase and the model sustains regular
phenomenology.
