This paper deals with analytic hypoellipticity in the sense of germs of several classes of linear partial differential equations. Recently, N. Hanges proved that the operator in \(\mathbb{R}^3\): \[ P=\partial^2_t+ t^2\Delta_x+ \partial^2_{\theta(x)},\;\partial_{\theta(x)}= x_1{\partial\over\partial x_2}- x_2{\partial\over\partial x_1}, \] is analytic hypoelliptic in the sense of germs at \(O\), but it fails to be analytic hypelliptic ``in the strong sense'' in any neighbourhood of \(O\) (i.e., there is no neighbourhood \(U\) of \(O\) such that for every open \(V\subset U\) and distribution \(u\) in \(U\), \(Pu\) analytic in \(V\) implies that \(u\) is analytic in \(V\)). The authors give at first a short \(L_2\) proof of this result and then generalize it for larger classes of operators with nonsymplectic characteristic varieties of the type \[ P_2= \Delta_t+ |t|^2\sum^k_{i,j=1} a_{ij} X_i X_j+ \sum^s_{i,j=1} b_{ij} Y_i Y_j. \] Thus, they prove that if \(Pu\) is analytic in some neighbourhood of \(O\) then so is \(u\) in a possibly smaller neighbourhood of \(O\). It is well known that \(P_2\) propagates the analytic wave front set along the Hamilton leaves of the characteristic manifold, but one may have analytic hypoellipticity in open sets of a certain geometry relative to \(P_2\).