TITLE: Why is every Hamiltonian system locally integrable?
QUESTION [5 upvotes]: It is common knowledge that every Hamiltonian system is locally integrable (away from singular points of the Hamiltonian), meaning that, in a neighborhood of each point of the $2n$-dimensional symplectic manifold on which the Hamiltonian vector field is defined, it is possible to find $n$ integrals of motion in involution.
In general these do not globally extend to give the compact Lagrangian fibration/foliation appearing in the Arnold-Liouville theorem, but here I want to focus on this purely local situation.
What is the simplest rigorous argument you can give me of this fact?

REPLY [9 votes]: For non-singular Hamiltonian systems, you can see it as a consequence of a slight generalization of the Darboux Theorem which is known as the Carathéodory--Jacobi--Lie Theorem (see, e.g. Libermann, Marle, Symplectic Geometry and Analytical Mechanics).

Carathéodory--Jacobi--Lie Theorem Let $(M,\omega)$ be a $2n$-dimensional symplectic manifold, with associated Poisson bracket $\{-,-\}$. Assume to have, for some $0\leq
 k\leq n$, smooth functions $f_1,\ldots,f_k$, defined on a neighborhood
  $U$ of a point $x$ in $M$, which are independent and
  Poisson-commuting, i.e.
$$df_1\wedge\ldots\wedge df_k\neq 0,$$
$$\{f_i,f_j\}=0,\text{ for all }0\leq i<j\leq k.$$
Then there exist smooth
  functions $f_{k+1},\ldots,f_n,g_1,\ldots,g_n$ defined an open neighborhood $V$ of $x$ in $U$ such that
$$\omega=\sum_{i=1}^n df_i\wedge dg_i.$$

I hope it helps.