  Yang-Mills gauge theory models on a cylinder coupled to external matter
charges provide powerful means to find and solve certain non-linear integrable
systems. We show that, depending on the choice of gauge group and matter
charges, such a Yang-Mills model is equivalent to trigonometric Calogero-Moser
systems and certain known spin generalizations thereof. Choosing a more general
ansatz for the matter charges allows us to obtain and solve novel integrable
systems. The key property we use to prove integrability and to solve these
systems is gauge invariance of the corresponding Yang-Mills model.
