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We show that various models of the elliptic Calogero-Moser systems areaccompanied with an isomonodromic system on a torus. The isomonodromic partneris a non-autonomous Hamiltonian system defined by the same Hamiltonian. Therole of the time variable is played by the modulus of the base torus. Asuitably chosen Lax pair (with an elliptic spectral parameter) of the ellipticCalogero-Moser system turns out to give a Lax representation of thenon-autonomous system as well. This Lax representation ensures that thenon-autonomous system describes isomonodromic deformations of a linear ordinarydifferential equation on the torus on which the spectral parameter of the Laxpair is defined. A particularly interesting example is the ``extended twisted$BC_\ell$ model'' recently introduced along with some other models by Bordnerand Sasaki, who remarked that this system is equivalent to Inozemtsev'sgeneralized elliptic Calogero-Moser system. We use the ``root type'' Lax pairdeveloped by Bordner et al. to formulate the associated isomonodromic system onthe torus.Comment: latex2e using amsfonts package, 50pages; (v2) typos corrected; (v3)  typos in (3.35), (3.46), (3.48) and (B.26) corrected; (v4) errors in  (1.7),(1.12),(3.46),(3.47) and (3.48) corrected; (v5) final version for  publication, errors in (2.31),(2.35),(3.12),(3.30),(3.45),(4.16) and (4.37)  correcte

Elliptic Calogero–Moser systems and isomonodromic deformations