2D Classical Integrable Field Theories from Hitchin Systems on $$\mathrm{SL}(NM,\mathbb C)$$-Bundles with Non-trivial Characteristic Classes

We review the construction of classical elliptic integrable systems on the $$\mathrm{SL}(N,\mathbb C)$$ Higgs bundles with non-trivial characteristic classes. The set of integrable models includes the Calogero–Moser system, its spin generalization, the Euler–Arnold top, and the Gaudin models. In some particular cases, mixed-type models known as the generalized models of interacting tops appear. Then we also review the field generalizations for these models on the $$(1\,{+}\,1)$$ -dimensional space–time. Finally, we propose a Maillet-type classical $$r$$ -matrix structure for the field analog of the generalized model of interacting tops corresponding to $$\mathrm{SL}$$ -bundles with non-trivial characteristic classes.