 Canonical completion of independent commuting functions

Independent functions $Q_1,\dots ,Q_n$ on a $2n$-dimensional symplectic manifold can locally form the position half of canonical coordinates exactly when $\{Q_i,Q_j\}=0$ throughout a neighbourhood. Their commuting Hamiltonian vector fields span the fibres of $Q$. Flow from a transverse section gives coordinates $(Q,t)$ with $\omega =\sum d{t}_i\wedge d{Q}_i+\beta (Q)$. Write $\beta =d\alpha$ locally and take $P_i=-t_i-{\alpha }_i(Q)$. Then $\omega =\sum d{Q}_i\wedge d{P}_i$.