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".
Best Answer
If your system is algebraic, then you bet! More generally, you can get abelian varieties as the fibers for many interesting integrable systems. Google the following for more: algebraic complete integrable Hamiltonian system, Calogero-Moser System, Hitchin System.
As for elliptic curves, they'll only pop out in low dimensional cases, because otherwise, the fibers have to have larger dimension.
As for the latter, it depends what you might want. I've seen the definition of integrable given by "can be solved by a sequence of quadratures" and in this terminology, you can check that an algebraic system you're always working with the global section of the theta function on the abelian variety, which is the unique (up to scaling) global section of the theta divisor on the abelian variety, which for an elliptic curve, is just the Weierstrass function.