  We consider topologically non-trivial Higgs bundles over elliptic curves with
marked points and construct corresponding integrable systems. In the case of
one marked point we call them the modified Calogero-Moser systems (MCM
systems). Their phase space has the same dimension as the phase space of the
standard CM systems with spin, but less number of particles and greater number
of spin variables. Topology of the holomorphic bundles are defined by their
characteristic classes. Such bundles occur if G has a non-trivial center, i.e.
classical simply-connected groups, $E_6$ and $E_7$. We define the conformal
version CG of G - an analog of GL(N) for SL(N), and relate the characteristic
classes with degrees of CG-bundles. Starting with these bundles we construct
Lax operators, quadratic Hamiltonians, define the phase spaces and the Poisson
structure using dynamical r-matrices.
  To describe the systems we use a special basis in the Lie algebras that
generalizes the basis of t'Hooft matrices for sl(N). We find that the MCM
systems contain the standard CM systems related to some (unbroken) subalgebras.
The configuration space of the CM particles is the moduli space of the
holomorphic bundles with non-trivial characteristic classes.
