Addition of lower order terms preserving almost hypoellipticity of polynomials

A linear differential operator $P(D)$ with constant coefficients is called almost hypoelliptic if all derivatives $P^{(\nu)}(\xi)$ of the characteristic polynomial $P(\xi)$ can be estimated above via $P(\xi)$. In this paper we describe the collection of lower order terms addition of which to an almost hypoelliptic operator $P(D)$ (polynomial $P(\xi)$) preserves its almost hypoellipticity and its strength.