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Calogero-Moser models and Toda models are well-known integrablemulti-particle dynamical systems based on root systems associated with Liealgebras. The relation between these two types of integrable models isinvestigated at the levels of the Hamiltonians and the Lax pairs. The Lax pairsof Calogero-Moser models are specified by the representations of the reflectiongroups, which are not the same as those of the corresponding Lie algebras. Thelatter specify the Lax pairs of Toda models. The Hamiltonians of the ellipticCalogero-Moser models tend to those of Toda models as one of the periods of theelliptic function goes to infinity, provided the dynamical variables areproperly shifted and the coupling constants are scaled. On the other hand mostof Calogero-Moser Lax pairs, for example, the root type Lax pairs, do not ahave consistent Toda model limit. The minimal type Lax pairs, which correspondsto the minimal representations of the Lie algebras, tend to the Lax pairs ofthe corresponding Toda models.Comment: LaTeX2e with amsfonts.sty, 33 pages, no figure

Calogero-Moser Models. IV: Limits to Toda Theory