A counterexample of Gevrey class to the uniqueness of the Cauchy problem

We shall here give a special case of a counterexample due to Alinhac and Baouendi [1,2] which is so explicit that we can show that it belongs to a Denjoy-Carleman class contained in every Gevrey class strictly larger than the analytic one.The example of [1,2] is the key to an example of nonuniqueness for the analytic nonlinear Cauchy problem due to Métivier [4], which we also recall adding such strong regularity properties.This may be of some interest since there exist uniqueness theorems which are valid for all Gevrey classes close to the analytic class but not for every Gevrey class (see Lerner [3]).