  Integrable $\sigma$-models, such as the principal chiral model,
${\mathbb{Z}}_T$-coset models for $T \in {\mathbb{Z}}_{\geq 2}$ and their
various integrable deformations, are examples of non-ultralocal integrable
field theories described by (cyclotomic) $r/s$-systems with twist function. In
this general setting, and when the Lie algebra ${\mathfrak{g}}$ underlying the
$r/s$-system is of classical type, we construct an infinite algebra of local
conserved charges in involution, extending the approach of Evans, Hassan,
MacKay and Mountain developed for the principal chiral model and symmetric
space $\sigma$-model. In the present context, the local charges are attached to
certain `regular' zeros of the twist function and have increasing degrees
related to the exponents of the untwisted affine Kac-Moody algebra
$\widehat{{\mathfrak{g}}}$ associated with ${\mathfrak{g}}$. The Hamiltonian
flows of these charges are shown to generate an infinite hierarchy of
compatible integrable equations.
