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

Local charges in involution and hierarchies in integrable sigma-models