Provide a complete smooth-category proof of the Eliasson normal form

Prove, in full generality in the smooth category, the Eliasson normal form theorem for integrable systems near non-degenerate singular points, establishing local symplectic coordinates and commuting quadratic normal forms that linearize the system, and, in particular, prove the existence of the local diffeomorphism g composing F with the model Q under the stated hypotheses (including the focus–focus and hyperbolic cases).

Background

The Eliasson normal form asserts that near any non-degenerate singular point of an integrable system (M,ω,F), there exist local symplectic coordinates in which the components of F Poisson-commute with a model of quadratic forms (elliptic, hyperbolic, focus–focus, and regular blocks). In the non-hyperbolic case, one further expects a local diffeomorphism g so that g∘F equals the quadratic model.

While the theorem is proved in various special cases (analytic case; fully elliptic; focus–focus in dimension four; elliptic/hyperbolic in dimension two), the author notes that a complete proof in the smooth category for all cases does not exist in the literature, highlighting a foundational gap.