  We study complexified elliptic Calogero-Moser integrable systems. We
determine the value of the potential at isolated extrema, as a function of the
modular parameter of the torus on which the integrable system lives. We
calculate the extrema for low rank B,C,D root systems using a mix of analytical
and numerical tools. For so(5) we find convincing evidence that the extrema
constitute a vector valued modular form for a congruence subgroup of the
modular group. For so(7) and so(8), the extrema split into two sets. One set
contains extrema that make up vector valued modular forms for congruence
subgroups, and a second set contains extrema that exhibit monodromies around
points in the interior of the fundamental domain. The former set can be
described analytically, while for the latter, we provide an analytic value for
the point of monodromy for so(8), as well as extensive numerical predictions
for the Fourier coefficients of the extrema. Our results on the extrema provide
a rationale for integrality properties observed in integrable models, and embed
these into the theory of vector valued modular forms. Moreover, using the data
we gather on the modularity of complexified integrable system extrema, we
analyse the massive vacua of mass deformed N=4 supersymmetric Yang-Mills
theories with low rank gauge group of type B,C and D. We map out their
transformation properties under the infrared electric-magnetic duality group as
well as under triality for N=1* with gauge algebra so(8). We find several
intriguing properties of the quantum gauge theories.
