An hamiltonian system with $n$ degree of freedom is said to be completely integrable when there exists an system $f_1,\ldots,f_n$ of first integrals mutually Poisson-commuting, such that $df_1(x),\ldots,df_n(x)$ are linearly independent for any $x$ in a dense subset.
Consequently any hamiltonian system with one degree of freedom, whose non fixed points constitute a dense subset, is completely integrable, (infact in this case just its hamilton function gives a maximal set of Poisson-commuting first integrals which are independent on a dense subset).
Now in a paper I read that any hamiltonian system with one degree of freedom is completely integrable, because there should exist always a smooth function which is Poisson-commuting with the Hamilton function and whose regular points are every-where dense.
I have some difficulty in understanding this statetement, so is there someone who could give me an hint?
My guess is that, given an Hamilton function $H$ on $(M,\omega)$ whose regular points constitute an open subset $U$, if it is possible to prove that there exists a smooth function $\delta$ vanishing on $U$ and whose regular points are dense in $M\setminus\overline{U}$, then we could take $H+\delta$.