Integrable dynamical system - relation to elliptic curves

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.

"The de-geometrisation of mathematical education and the divorce from physics sever these ties. For example, not only students but also modern algebro-geometers on the whole do not know about the Jacobi fact mentioned here: an elliptic integral of first kind expresses the time of motion along an elliptic phase curve in the corresponding Hamiltonian system. "

From A.I.Arnold, here: http://pauli.uni-muenster.de/~munsteg/arnold.html Definitely I should learn more in this area....