Completely Integrable Contact Hamiltonian Systems and Toric Contact Structures on S ² ×S ³

I begin by giving a general discussion of completely integrable Hamiltonian systems in the setting of contact geometry. We then pass to the particular case of toric contact structures on the manifold S 2 S 3 . In particular we give a complete solution to the contact equivalence problem for a class of toric contact structures, Y p,q , discovered by physicists by showing that Y p,q and Y p ,q are inequivalent as contact structures if and only if p = p .