  We show that various models of the elliptic Calogero-Moser systems are
accompanied with an isomonodromic system on a torus. The isomonodromic partner
is a non-autonomous Hamiltonian system defined by the same Hamiltonian. The
role of the time variable is played by the modulus of the base torus. A
suitably chosen Lax pair (with an elliptic spectral parameter) of the elliptic
Calogero-Moser system turns out to give a Lax representation of the
non-autonomous system as well. This Lax representation ensures that the
non-autonomous system describes isomonodromic deformations of a linear ordinary
differential equation on the torus on which the spectral parameter of the Lax
pair is defined. A particularly interesting example is the ``extended twisted
$BC_\ell$ model'' recently introduced along with some other models by Bordner
and Sasaki, who remarked that this system is equivalent to Inozemtsev's
generalized elliptic Calogero-Moser system. We use the ``root type'' Lax pair
developed by Bordner et al. to formulate the associated isomonodromic system on
the torus.
