  We continue our study of nonperturbative superpotentials of four-dimensional
N=2 supersymmetric gauge theories with gauge group U(N) on R^3 x S^1, broken to
N=1 due to a classical superpotential. In a previous paper, hep-th/0304061, we
discussed how the low-energy quantum superpotential can be obtained by
substituting the Lax matrix of the underlying integrable system directly into
the classical superpotential. In this paper we prove algebraically that this
recipe yields the correct factorization of the Seiberg-Witten curves, which is
an important check of the conjecture. We will also give an independent proof
using the algebraic-geometrical interpretation of the underlying integrable
system.
