  Calogero-Moser models and Toda models are well-known integrable
multi-particle dynamical systems based on root systems associated with Lie
algebras. The relation between these two types of integrable models is
investigated at the levels of the Hamiltonians and the Lax pairs. The Lax pairs
of Calogero-Moser models are specified by the representations of the reflection
groups, which are not the same as those of the corresponding Lie algebras. The
latter specify the Lax pairs of Toda models. The Hamiltonians of the elliptic
Calogero-Moser models tend to those of Toda models as one of the periods of the
elliptic function goes to infinity, provided the dynamical variables are
properly shifted and the coupling constants are scaled. On the other hand most
of Calogero-Moser Lax pairs, for example, the root type Lax pairs, do not a
have consistent Toda model limit. The minimal type Lax pairs, which corresponds
to the minimal representations of the Lie algebras, tend to the Lax pairs of
the corresponding Toda models.
