Spin Calogero-Moser models on symmetric spaces

In this paper we construct and prove superintegrability of spin Calogero-Moser type systems on symplectic leaves K 1 ∖ T ∗ G / 2 K_1\backslash T^*G/K_2 where comma 2 subset-of G"> , ⊂<!-- ⊂ encoding="application/x-tex">K_1,\space K_2\subset G are subgroups. We call them two-sided systems. One important such correspond to equals K"> = encoding="application/x-tex">K_1=K_2=K encoding="application/x-tex">K is a subgroup fixed points Chevalley involution alttext="theta colon right-arrow θ<!-- θ <mml:mo>: →<!-- → encoding="application/x-tex">\theta : G\to . The other series examples come from pair times ×<!-- × encoding="application/x-tex">G\subset G\times with the diagonal embedding. explicitly describe corresponding rank one when S L Subscript n"> S L n encoding="application/x-tex">G=SL_n