Relative Moser theorem

The closed form $\alpha ={\omega }_1-{\omega }_0$ vanishes as a tensor along $S$. By [F1], after shrinking there is a one-form $\sigma$ with $d\sigma =\alpha$ and vanishing first jet along $S$. Solve ${\iota }_{X_t}{\omega }_t=-\sigma$; [F2] gives a smooth $X_t$, and nondegeneracy gives both $X_t{\mathrm{\mid }}_S=0$ and vanishing first jet there.

By [F3], around each point of $S$ there is a neighbourhood whose trajectories exist through the compact time interval after finitely many local continuations. Their union contains $S$; uniqueness glues the evolutions and makes the time-one map a diffeomorphism onto its open image. Since $X_t{\mathrm{\mid }}_S=0$, every point of $S$ is fixed.

For the evolution ${\varphi }_t$, [F2] yields $\frac{d}{dt}({\varphi }_t^{\ast }{\omega }_t)=0$. Thus ${\varphi }_1^{\ast }{\omega }_1={\omega }_0$ on a possibly smaller source neighbourhood. Set $U_0$ equal to that domain and $U_1={\varphi }_1(U_0)$.