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

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.