  We define the quasi-compact Higgs $G^{\mathbb C}$-bundles over singular
curves introduced in our previous paper for the Lie group SL($N$). The
quasi-compact structure means that the automorphism groups of the bundles are
reduced to the maximal compact subgroups of $G^{\mathbb C}$ at marked points of
the curves. We demonstrate that in particular cases this construction leads to
the classical integrable systems of Hitchin type. The examples of the systems
are analogues of the classical Calogero-Sutherland systems related to a simple
complex Lie group $G^{\mathbb C}$ with two types of interacting spin variables.
These type models were introduced previously by Feher and Pusztai. We construct
the Lax operators of the systems as the Higgs fields defined over a singular
rational curve. We also construct hierarchy of independent integrals of motion.
Then we pass to a fixed point set of real involution related to one of the
complex structures on the moduli space of the Higgs bundles. We prove that the
number of independent integrals of motion is equal to the half of dimension of
the fixed point set. The latter is a phase space of a real completely
integrable system. We construct the classical $r$-matrix depending on the
spectral parameter on a real singular curve, and in this way prove the complete
integrability of the system. We present three equivalent descriptions of the
system and establish their equivalence.
