Geometric results for hyperbolic operators with spectral transition of the Hamilton map

In this paper we study a class of non-effectively hyperbolic operators vanishing of order 2 on a manifold, on a sub-region of which the spectral structure of the Hamilton map changes type. Suitable normal symplectic coordinates are found together with an analysis of the Hamilton system associated to the principal symbol and a factorization result, preparing the operator for a microlocal energy estimate, is finally proven.