2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/99075The recent example of Hanges: $P = \partial_t^2 + t^2Δ_x + \partial^2_{θ(x)}$ in $R^3$ is analytic hypoelliptic in the sense of germs but not in the strong sense in any neighborhood of the origin. And its characteristic variety is non-symplectic. We give a purely $L^2,$ and hence quite flexible, proof of this result and generalizations, and link it to, and contrast it with, the celebrated Baouendi-Goulaouic operator. We point out that the results are consistent with the conjecture of Treves.11 pagesAnalysis of PDEsComplex VariablesSymplectic Geometry35H10; 35N15text