I'm wondering if there is some "nice" version of Darboux's theorem that can be applied in the case of a symplectisation of a contact manifold $(X,\alpha)$ with the canonical symplectic form. i.e. $(\Bbb R \times X, d(e^t\alpha))$, where $t$ is the $\Bbb R$-parameter.
More specifically, I'd like to know if we can assume that the first 2 coordinates in a Darboux neighbourhood are given by the Reeb vector field and the canonical Liouville vector field $\partial_t$.
I tried looking around but wasn't really able to find anything