This question is related to a [previous one](https://mathoverflow.net/questions/333245/practical-example-of-hamiltonian-reduction); now I better understand the problem and I can more clearly state what is the question.


**Background**


I refer to the following concepts:


Liouville integrability: a Hamiltonian with $n$ degrees of freedom has $n$ independent integrals of motion in involution; we know that the Hamiltonian can be brought to the form $H(p\_1, \dots, p\_n)$ (i.e. independent on the $q$s) by a canonical transformation.


Poincare' reduction: the Hamiltonian has **one** integral of motion (just to simplify!); we can bring the Hamiltonian to the form $H(p\_1, q\_1, \dots, p\_{n-1}, q\_{n-1}, p\_n)$ (i.e. we remove the dependence on one of the $q$s, $q\_n$) by means of a canonical transformation. 


**Global versus local**


There are examples of both operations (Liouville integration or Poincare' reduction) that are performed globally. 


We also know that the Liouville integration is guaranteed to be feasible globally under a non-degeneracy condition (Arnold-Liouville theorem). Else, if the non-degenerasy condition is not met, the integrability is guaranteed only locally.


On the other hand, the feasibility of Poincare' reduction is discussed only locally (for what I've seen, e.g. Arnold, "Mathematical Aspects of Classical and Celestial Mechanics", proposition 3.2).


**What I'm asking**


I would like to know if there is any result which guarantees the global feasibility of the Poincare' reduction. This could possibly include a non-degeneracy condition (as in the case of Liouville's integration) or a wider range of forms of the reduced Hamiltonian.



Global reduction of Hamiltonian with an integral of motion (Poincare' reduction)
It's probably easier to think about this geometrically: suppose we have a function $F:M\to\mathbb{R}$ (where $M$ is your phase space) such that $\lbrace H, F\rbrace = 0$, and also the level sets $F^{-1}(c)$ are all (embedded) submanifolds of $M$. Then we can take $P\_n=F$, and from that construct canonical coordinates $Q\_i, P\_i$. The fact that $\lbrace H, F\rbrace = 0$ means that $H$ is invariant under the flow produced by taking $F=P\_n$ as the Hamiltonian. This flow maps $Q\_n$ to $Q\_n+t$, and preserves all other coordinates. Hence $H$ is independent of $Q\_n$.


The requirement $dF \ne 0$ everywhere ensures that $F^{-1}(c)$ is actually an embedded submanifold for each $c$ (this is often referred to as the [Submersion Theorem](https://en.wikipedia.org/wiki/Submersion_(mathematics)#Submersion_theorem)). I believe the reason your counterexample (linked in the chatroom) fails is that it does not satisfy this condition: the Henon-Heiles Hamiltonian $H$ has several points where $dH = 0$, even if you exclude the point $(q\_1,q\_2,p\_1,p\_2)=(0,0,0,0)$ (incidentally, these are the equilibrium points of the system).


Edit: I realise now I'm not really addressing your actual question, but more the counterexample (linked to in the chatroom). One issue with the discussion above is that the new canonical coordinates $Q\_i, P\_i$ are only defined locally in general, and you're correct in asserting that what you call Poincare reduction is really only a local construction. Marsden-Weinstein reduction (which is a global construction) would be applicable in the case that the space of trajectories $F^{-1}(c)/\mathbb{R}$ of the Hamiltonian $F$ in $F^{-1}(c)$ can be given a smooth structure such that $\pi:F^{-1}(c)\to F^{-1}(c)/\mathbb{R}$ is a submersion. One criterion for this is that the $\mathbb{R}$-action on $M$ determined by $F$ is [proper](http://mathworld.wolfram.com/ProperGroupAction.html), but I'm not sure how useful this would be for you (sorry).


