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.

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