  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 $S^2\times S^3$. 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
$p\neq p'$.
