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?