# Integrability of the tautological 1-form

• I
MathNeophyte
TL;DR Summary
Trying to understand under what conditions the tautological/Liouville one form is integrable in the sense of having $n$ Poisson commuting integrals.
Apologies for potentially being imprecise and clunky, but I'm trying understand integrability of the following Hamiltonian
$$H(x,p)=\langle p,f(x) \rangle$$
on a 2n dimensional vector space
$$T^{\ast}\mathcal{M} =\mathbb{R}^{2n}.$$
Clearly this is just the 1-form $$\theta_{(x,p)} = \sum_{i}{p_i{}dq^{i}}.$$
I assume that anywhere along the flow of x given some x0
$$x(\tau) = \Phi(x_{0},\tau)$$ the Hamiltonian is non-degenerate i.e. the Hessian of the Hamiltonian is full rank.

My questions are:
1. Is H(x,p) Liouville integrable along the orbits of x given the above non-degeneracy assumption? I.e. Does this Hamiltonian have n, non-constant integrals that are commuting with respect to the cannonical Poisson bracket?
2. How does this relate to the differential form version of the Frobenius theorem?
I would appreciate any and all pointers, this clearly isn't my domain of expertise but it is a question that arose in some work on dynamical systems.

jbergman