Integrability, hyperbolic flows and the Birkhoff normal form

We prove that a Hamiltonian $p\in C^\infty(T^*{\bf R}^n)$ is locally integrable near a non-degenerate critical point $\rho_0$ of the energy, provided that the fundamental matrix at $\rho_0$ has no purely imaginary eigenvalues. This is done by using Birkhoff normal forms, which turn out to be convergent in the $C^\infty$ sense. We also give versions of the Lewis-Sternberg normal form near a hyperbolic fixed point of a canonical transformation. Then we investigate the almost holomorphic case.