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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

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This is not possible. Let $\omega = d(e^t\alpha)$, and suppose $(x^1,\dots,x^n,y^1,\dots,y^n)$ are any Darboux coordinates for $\omega$. Then the vector fields $\partial _{x^i}$ and $\partial_{y^i}$ are all symplectic (or equivalently locally Hamiltonian), which means that the Lie derivative of $\omega$ with respect to each of them is zero. But $\partial_t$ is not symplectic, because $$ \mathscr L_{\partial_t}\omega = d(i_{\partial_t}\omega) = d(e^t\alpha) = \omega. $$

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