Completely Integrable Bi-Hamiltonian Systems

We study the geometry of completely integrable bi-Hamiltonian systems, and in particular, the existence of a bi-Hamiltonian structure for a com-pletely integrable Hamiltonian system. We show that under some natural hypothesis, such a structure exists in a neighborhood of an invariant torus if, and only if, the graph of the Hamiltonian function is a hypersurface of translation, relative to the affine structure determined by the action variables. This generalizes a result of Brouzet for dimension four. KEY WORDS: Bi-Hamiltonian system; completely integrable system. The study of completely integrable Hamiltonian systems, i.e., systems admitting a complete sequence of first integrals, started with the pionneering work of Liouville (1855) on finding local solutions by quadratures. We have now a complete picture of the geometry of such systems, which in its modern presentation is due to Arnol’d (1988). A major flaw in th