Projectable contact vector fields of projective cotangent bundle

Question

Consider the projective cotangent bundle $PT^\star M$ with the contact structure inherited from the kernel of the Liouville form of $T^\star M$. A contact vector field $X$ (infinitesimal contactomorphism) on $PT^\star M$ can locally be the Reeb field $E$ of a contact form $\omega$, and there is always a coordinate system on $PT^\star M$ such that $\omega = dx^0 - y_i dx^i$ locally ($i=1..n$; Einstein summation convention). In this coordinate system, $E$ is $\partial/\partial x^0$. Now, $x^0$ and $x^i$ can be pulled-back coordinates from the base $M$ to the bundle, and therefore $E$ projects to the vector field $\partial/\partial x^0$ on $M$. Do these show that any contact vector field $X$ on $PT^\star M$ is always projectable locally to a vector field $\pi_\star X$ on $M$ (by the projection $\pi:PT^\star M \rightarrow M$), or am I confused about something here?

P.S.: Why do I think $x^0$ and $x^i$ could be pulled-back from $M$? It is because $T^\star M$ can be given coordinates $x^0$, $x^i$, $p_0$, $p_i$, where $x^0$ and $x^i$ are on $M$, and the restriction of $dx^0 + p_i/p_0 dx^i$ ($p_0 \neq 0$) to $PT^\star M$, obtained from the Liouville form on $T^\star M$, is a contact form.

Answer 1

Here is a counterexample.

$\DeclareMathOperator{\d}{d} \newcommand{\dx}{\d \!}$ Consider $T^2\times S^1$, which is naturally isomorphic as a circle bundle to the projective cotangent bundle $PT^*(T^2)$ of the $2$-torus $T^2$. The canonical contact form of $PT^*(T^2)$ reads $\cos\theta \dx x + \sin \theta \dx y$ (it is claimed in this article, for instance), whose Reeb vector field $\cos \theta\, \partial_x + \sin\theta\, \partial_y$ is tautologically a contact vector field, but does not project onto the base $T^2$.

Answer 2

As pointed out by @Didier, $x^0$ as a Darboux coordinate is not necessarily pulled back from the base $M$. The point that I was missing is that not all contact forms representing the same contact structure on $PT^\star M$ (inherited from the kernel of the Liouville form of $T^\star M$) have to be in the form $dx^0 - y_i dx^i$ where $x^0$ and $x^i$ are pulled back from $M$. A form $h(dx^0 - y_i dx^i)$, where $h=h(x^0, x^i, y_i)$, represents the same contact structure, and the Darboux coordinates of the latter do not have to be pulled-back in general.