This question is related to a previous one; 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.

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