On the sharp Gevrey regularity for a generalization of the Métivier operator

We prove a sharp Gevrey hypoellipticity for the operator $$\begin{aligned} D_{x}^{2}+\left( x^{2n+1}D_{y}\right) ^{2}+\left( x^{n}y^{m}D_{y}\right) ^{2}, \end{aligned}$$ in $$\Omega $$ open neighborhood of origin $${\mathbb {R}}^{2}$$ , where n and m are positive integers. The is non trivial generalization Métivier studied (C R Acad Sci Paris 292:401–404, 1981). However it has symplectic characteristic manifold stratum according to Poisson–Treves stratification. According Treves conjecture turns out not be analytic hypoelliptic.