As always, I'm not sure if I'm about to ask a very stupid question, and I apologise if that is the case.
Most systems from physics come from classical Hamiltonians, defined on the phase space of some manifold. Integrable systems happen when such a Hamiltonian is invariant under a sufficent number of transformations. On the other hand, an integrable system in symplectic geometry is defined as a $2n$-dimensional symplectic manifold, together with $n$ almost everywhere linearly indepedent functions which Poisson commute. The compact level sets of regular values of these functions are tori, to which the Hamiltonian vector fields of the functions are tangent. To connect with the "from physics" case, the system evolves along level sets of quantities conserved by the transfomations. So far so good.
It would be good to know what can be said, if anything, about the embedding of the configuration space, or base manifold, into the phase space of the integrable system with Arnold-Liouville coordinates. To maybe give an idea of what I mean, looking at the classical space of the pendulum, the embedding of the zero section of the tangent bundle is transverse to the invariant tori and contains both critical points. Is there some hope that there are things that can be said about the embedding of the zero section of the tangent bundle into the integrable system and it's relationship to the invariant structures in it? Maybe things can be said if the hamiltonian is classical? Can we say anything about special integrable systems that are also globally cotangent bundles of some manifold together with a classical hamiltonians? (i.e. sum of kinetic and potential energy depending only on the base coordinate).
