  This paper is a continuation of our previous paper \cite{LOSZ}. For simple
complex Lie groups with non-trivial center, i.e. classical simply-connected
groups, $E_6$ and $E_7$ we consider elliptic Modified Calogero-Moser systems
corresponding to the Higgs bundles with an arbitrary characteristic class.
These systems are generalization of the classical Calogero-Moser (CM) systems
related to a simple Lie groups and contain CM systems related to some
(unbroken) subalgebras. For all algebras we construct a special basis,
corresponding to non-trivial characteristic classes, the explicit forms of Lax
operators and Hamiltonians.
