# Connection between integrable systems and group actions

An integrable system can be defined as a symplectic manifold together with the maxiumum possible number of Poisson commuting functions on the manifold which are almost everywhere independent. By the Liouville-Arnold theorem such manifolds have an intimate connection with the idea of toric fibrations or toric symmetries as, about ''regular points'' (where the Poisson commuting functions are independent), the symplectic form can be written in action angle coordinites and the manifold admits a fibration by Lagrangian tori, with the flow of the Hamiltonian vector fields occuring on these tori. One can find a ''momentum map'' for such a system about a regular orbit, which is a particular instance of symplectic reduction and one can then, for example, examine the dynamics on the reduced space.

One also hears of systems, such as certain types of rigid body motion, being described as ''integrable'', though this seems to have a slightly different flavour to it. In the case of the free rigid body, for example, the system is reduced by the group $SO(3)$. We can take a ''momentum map'' to the Lie algebra $\mathbb{so(3)}^*$ and look at the reduced dynamics on this Lie algebra. What is the connection between this type of integrability and Liouville-Arnold integrability? Will a Hamiltonian group action by a group of the maximal possible dimension produce an integrable system in the Arnold-Liouville sense, or are these totally seperate concepts? If there is an associated Arnold-Liouville system, what is the relation between the two momentum maps (the toric one and the one to $\mathbb{so(3)}^*$)?