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?



Why is every Hamiltonian system locally integrable?
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.


