A non-symplectic KAM phenomenon for singular holomorphic vector fields (Q2721627)

The author performs an interesting investigation of the germs of holomorphic vector fields at the origin of \(\mathbb C^{n}\). If such a germ \(X\) is a perturbation of a singular completely integrable system and its linear part admits non-trivial polynomial first integrals, then it admits a lot of invariant analytic subsets in a neighbourhood of \(0\) and, despite the fact that \(X\) is not holomorphically normalizable, it has a holomorphic normal form in a neighbourhood of the origin. An open question is thus answered.