Integrable Systems and surfaces

The simplest possible integrable Hamiltonian system lies in two dimensions, and is taught in a first course on o.d.e., as ``exact systems". One can of course then take products of these systems, and obtain systems in higher dimensions. It turns out that most of the integrable systems commonly studied (tops, finite gap cases of KdV, etc) are in fact of this type, once they are suitably interpreted.