From seminar on kdV equation I know that for integrable dynamical system its trajectory in phase space lays on tori. In wikipedia article You may read (http://en.wikipedia.org/wiki/Integrable_system):
When a finite dimensional Hamiltonian system is completely integrable in the Liouville sense, and the energy level sets are compact, the flows are complete, and the leaves of the invariant foliation are tori. There then exist, as mentioned above, special sets of canonical coordinates on the phase space known as action-angle variables, such that the invariant tori are the joint level sets of the action variables. These thus provide a complete set of invariants of the Hamiltonian flow (constants of motion), and the angle variables are the natural periodic coordinates on the torus. The motion on the invariant tori, expressed in terms of these canonical coordinates, is linear in the angle variables.
As I also know that elliptic curve is in fact some kind of tori, then there natural question arises: Are tori for quasi-periodic motion in action-angle variables of some dynamical systems related in any way to algebraic structure like elliptic curve? Maybe some small dynamical systems and some elliptic curves are related in some way?
The most interesting in this matter is for me the size of space of elliptic functions: its quite small, every elliptic curve is rational function of Weiestrass function, and its derivative. Has this property any analogy in integrable dynamical systems theory?
As isomorphic elliptic curves shares some invariants, it is also interesting it they have any "dynamical meaning".