  We study exact effective superpotentials of four-dimensional {\cal N} = 2
supersymmetric gauge theories with gauge group U(N) and various amounts of
fundamental matter on R^3 x S^1, broken to {\cal N} = 1 by turning on a
classical superpotential for the adjoint scalar. On general grounds these
superpotentials can easily be constructed once we identify a suitable set of
coordinates on the moduli space of the gauge theory. These coordinates have
been conjectured to be the phase space variables of the classical integrable
system which underlies the {\cal N} = 2 gauge theory. For the gauge theory
under study these integrable systems are degenerations of the classical,
inhomogeneous, periodic SL(2,C) spin chain. Ambiguities in the degeneration
provide multiple coordinate patches on the gauge theory moduli space. By
studying the vacua of the superpotentials in several examples we find that the
spin chain provides coordinate patches that parametrize holomorphically the
part of the gauge theory moduli space which is connected to the electric (as
opposed to magnetic or baryonic) Higgs and Coulomb branch vacua. The baryonic
branch root is on the edge of some coordinate patches. As a product of our
analysis all maximally confining (non-baryonic) Seiberg-Witten curve
factorizations for N_f \leq N_c are obtained, explicit up to one constraint for
equal mass flavors and up to two constraints for unequal mass flavors. Gauge
theory addition and multiplication maps are shown to have a natural counterpart
in this construction. Furthermore it is shown how to integrate in the meson
fields in this formulation in order to obtain three and four dimensional
Affleck-Dine-Seiberg-like superpotentials.
