> **Disclaimer**: The original question consisted of two parts. The first one
> has been answered negatively (see
> below the answers of Sam Lisi and
> Alejandro). It remains the second one.

**Background**  
I am reading about the energy-period relation for Hamiltonian Systems.  
In Weinstein's formulation (cf. Abraham, Marsden, Foundations of Mechanics 2nd Ed, page 198) this relation amounts to:
>$(\ast)$ Given an Hamiltonian system $(M,\omega, X_H)$, let be $\Phi$ the flow of $X_H$ and $\text{per}:=\{(t,x)\in\mathbb R\times M\mid\Phi(t,x)=x\}.$  
If $N$ is a smooth submanifold contained in $\text{per},$ then $\left.dt\wedge dH\right|_N=0,$ i.e. $t=t(H)$ on $N,$ (the period depends only on the energy.)

**Question**  
In Guillemin, Stenberg, Geometric Asymptotics, between pages [170-171][2], I have additionally found that, when all integral curves of $X_H$ are periodic, we can take $N=\text{per}$ in $(\ast),$ which should mean that in such a case $\text{per}$ is a smooth submanifold of $\mathbb R\times M.$  

In order to justify this last point I was wondering myself:

> 1. If $X$ is a non singular vector field on $M,$ all of whose integral curves are periodic, and $\tau(p)$ denotes the period of the orbit through $p,$ then $\tau:M\to\mathbb R$ is smooth? 
> 2. otherwise, how to prove that in such a case $\text{per}$ is a submanifold?

**What I have tried about point 2**  
Probably I am missing something because my guess is that if there were a principal bundle structure $(M,p,X,\mathbb S^1)$ such that the $\mathbb S^1$-orbits are the trajectories of $X$ then the period $\tau:M\to\mathbb R$ should be smooth because of the relation $\zeta=\tau X_H,$ where $\zeta$ is the infinitesimal generator of the action.  
But I don't know how to proceed without this additional hypothesis.  

**Edit1 (After Sebastian's answer about point 1):** As illustration of my difficulties with point 1, I imagine that $M$ is the Moebius band $[0,1]\times\mathbb R/\sim$ and $X=\frac{\partial}{\partial x}$ then the period is $$\tau([(x,y)]_{\sim})=\begin{cases}1&\text{if }y=0\\\2&\text{if }y\neq 0\end{cases}$$
![alt text][3]



  [1]: http://math.stackexchange.com/questions/179182/about-the-energy-period-relation-for-hamiltonian-systems
  [2]: http://books.google.it/books?id=58PgdwJzirUC&lpg=PP1&ots=GDIGlP2eaz&dq=guillemin%20sternberg%20Geometric%20Asymptotics&pg=PA170#v=onepage&q&f=false
  [3]: https://upload.wikimedia.org/wikipedia/commons/thumb/1/19/Moebius_strip.svg/267px-Moebius_strip.svg.png