Reeb vector field

Remarks

Existence and uniqueness: $d\alpha$ has rank $2n$ at each point, so its kernel is a line, and $\alpha$ is nonzero on this line because $\alpha \wedge (d\alpha)^n \ne 0$. The flow of $R_\alpha$ preserves $\alpha$, by Cartan's formula $\mathcal{L}_{R_\alpha} \alpha = d(\alpha(R_\alpha)) + \iota_{R_\alpha} d\alpha = 0$. The Reeb field depends on $\alpha$, not only on the contact structure: replacing $\alpha$ by $h\alpha$ changes it in general.

Example: on $\R^3$ with $\alpha = dz - y\,dx$, one has $d\alpha = dx \wedge dy$, and $R_\alpha = \partial_z$, since $\alpha(\partial_z) = 1$ and $\iota_{\partial_z}(dx \wedge dy) = 0$. On $S^3 \subset \C^2$ with $\alpha = \sum_j (x_j\, dy_j - y_j\, dx_j)$, the Reeb field is $R_\alpha = \sum_j (x_j \partial_{y_j} - y_j \partial_{x_j})$, whose flow is $z \mapsto e^{it} z$.

Non-example: on $\R^3$ with $\alpha = dz - y\,dx$, the field $\partial_y$ satisfies $\iota_{\partial_y} d\alpha = -dx \ne 0$ and $\alpha(\partial_y) = 0$, so it is not the Reeb field.