  We develop a new, systematic approach towards studying the integrability of
the ordinary Calogero-Moser-Sutherland models as well as the elliptic Calogero
models associated with arbitrary (semi-)simple Lie algebras and with symmetric
pairs of Lie algebras. It is based on the introduction of a function F, defined
on the relevant root system and with values in the respective Cartan
subalgebra, satisfying a certain set of combinatoric identities that ensure, in
one stroke, the existence of a Lax representation and of a dynamical R-matrix,
given by completely explicit formulas. It is shown that among the simple Lie
algebras, only those belonging to the A-series admit such a function F, whereas
the AIII-series of symmetric pairs of Lie algebras, corresponding to the
complex Grassmannians SU(p,q)/S(U(p) x U(q)), allows non-trivial solutions when
|p-q| <= 1. Apart from reproducing all presently known dynamical R-matrices for
Calogero models, our approach provides new ones, namely for the ordinary models
when |p-q| = 1 and for the elliptic models when |p-q| = 1 or p = q.
