  We prove the existence of novel, nonminimal and irreducible solutions to the
(self-dual) Ginzburg-Landau equations on closed surfaces. To our knowledge
these are the first such examples on nontrivial line bundles, that is, with
nonzero total magnetic flux. Our method works with the 2-dimensional,
critically coupled Ginzburg-Landau theory and uses the topology of the moduli
space. The method is nonconstructive, but works for all values of the remaining
coupling constant. We also prove the instability of these solutions.
