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We define the quasi-compact Higgs $G^{\mathbb C}$-bundles over singularcurves introduced in our previous paper for the Lie group SL($N$). Thequasi-compact structure means that the automorphism groups of the bundles arereduced to the maximal compact subgroups of $G^{\mathbb C}$ at marked points ofthe curves. We demonstrate that in particular cases this construction leads tothe classical integrable systems of Hitchin type. The examples of the systemsare analogues of the classical Calogero-Sutherland systems related to a simplecomplex Lie group $G^{\mathbb C}$ with two types of interacting spin variables.These type models were introduced previously by Feher and Pusztai. We constructthe Lax operators of the systems as the Higgs fields defined over a singularrational 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 thecomplex structures on the moduli space of the Higgs bundles. We prove that thenumber of independent integrals of motion is equal to the half of dimension ofthe fixed point set. The latter is a phase space of a real completelyintegrable system. We construct the classical $r$-matrix depending on thespectral parameter on a real singular curve, and in this way prove the completeintegrability of the system. We present three equivalent descriptions of thesystem and establish their equivalence.

Quasi-compact Higgs bundles and Calogero-Sutherland systems with two types spins