This is once again from Hansjorg Geiges' introduction to contact topology. In Gray Stability Theorem in $\S2.2$ asserts that one can achieve stability of contact structures. However, one can't in general achieve stability of contact forms. An example following Gray Stability Theorem, shows that one obstruction is the presence of closed orbits.
In Darboux's theorem $\S2.5$ he chooses a neighbourhood about the origin small enough that there are no closed orbits there.
I have two questions.
Why does such a neighbourhood exist? (For instance why can't the closed orbits be shrinking wedge of circles?)
Why does mere absence of closed orbits ensure that one can solve stability of forms in that chosen neighbourhood?
