  We continue our study of the N=1* supersymmetric gauge theory and its
relation to elliptic integrable systems. Upon compactification on a circle, we
show that the semi-classical analysis of the massless and massive vacua depends
on the classification of nilpotent orbits, as well as on the conjugacy classes
of the component group of their centralizer. We demonstrate that
semi-classically massless vacua can be lifted by Wilson lines in unbroken
discrete gauge groups. The pseudo-Levi subalgebras that play a classifying role
in the nilpotent orbit theory are also key in defining generalized Inozemtsev
limits of (twisted) elliptic integrable systems. We illustrate our analysis in
the N=1* theories with gauge algebras su(3), su(4), so(5) and for the
exceptional gauge algebra G2. We map out modular duality diagrams of the
massive and massless vacua. Moreover, we provide an analytic description of the
branches of massless vacua in the case of the su(3) and the so(5) theory. The
description of these branches in terms of the complexified Wilson lines on the
circle invokes the Eichler-Zagier technique for inverting the elliptic
Weierstrass function. After fine-tuning the coupling to elliptic points of
order three, we identify the Argyres-Douglas singularities of the su(3) theory.
