  Ginzburg-Landau fields are the solutions of the Ginzburg-Landau equations
which depend on two positive parameters, $\alpha$ and $\beta$. We give
conditions on $\alpha$ and $\beta$ for the existence of irreducible solutions
of these equations. Our results hold for arbitrary compact, oriented,
Riemannian 2-manifolds (for example, bounded domains in $\mathbb{R}^2$,
spheres, tori, etc.) with de Gennes-Neumann boundary conditions. We also prove
that, for each such manifold and all positive $\alpha$ and $\beta$, the
Ginzburg-Landau free energy is a Palais-Smale function on the space of gauge
equivalence classes, Ginzburg-Landau fields exist for only a finite set of
energy values, and the moduli space of Ginzburg-Landau fields is compact.
