  We have established a prescription for the calculation of analytical vortex
solutions in the context of generalized Maxwell-Higgs models whose overall
dynamics is controlled by two positive functions of the scalar field. We have
also determined a natural constraint between these functions and the Higgs
potential allowing the existence of axially symmetric
Bogomol'nyi-Prasad-Sommerfield (BPS) solutions possessing finite energy.
Furthermore, when the generalizing functions are chosen suitably, the
nonstandard BPS equations can be solved exactly. We have studied some examples,
comparing them with the usual Abrikosov-Nielsen-Olesen (ANO) solution. The
overall conclusion is that the analytical self-dual vortices are well-behaved
in all relevant sectors, strongly supporting the generalized models they belong
themselves. In particular, our results mimic well-known properties of the usual
(numerical) configurations, as localized energy density, while contributing to
the understanding of topological solitons and their description by means of
analytical methods.
