  The canonical structure of classical non-linear sigma models on Riemannian
symmetric spaces, which constitute the most general class of classical
non-linear sigma models known to be integrable, is shown to be governed by a
fundamental Poisson bracket relation that fits into the $r$-$s$-matrix
formalism for non-ultralocal integrable models first discussed by Maillet. The
matrices $r$ and $s$ are computed explicitly and, being field dependent,
satisfy fundamental Poisson bracket relations of their own, which can be
expressed in terms of a new numerical matrix~$c$. It is proposed that all these
Poisson brackets taken together are representation conditions for a new kind of
algebra which, for this class of models, replaces the classical Yang-Baxter
algebra governing the canonical structure of ultralocal models. The Poisson
brackets for the transition matrices are also computed, and the notorious
regularization problem associated with the definition of the Poisson brackets
for the monodromy matrices is discussed.
