Analytic surjectivity of all linear partial differential operators with constant coefficients on \({\mathbb{R}}^ 2\) and existence of a counterexample for the case of operators in \({\mathbb{R}}^ 3\) have been proved. In this paper, the authors prove the surjectivity of the operator \(D_ 2+ix_ 2^{2k}D_ 1\), which is an operator with variable coefficients, from the Gevrey space \({\mathcal E}^{\{s\}}({\mathbb{R}}^ 2)\) into itself and its non-surjectivity from \({\mathcal E}^{\{s\}}({\mathbb{R}}^ 3)\) to \({\mathcal E}^{\{s\}}({\mathbb{R}}^ 3)\).