Images of derivations of polynomial algebras with divergence zero

In this paper, we show that the image of a derivation D of k[x,y] with divergence zero is not necessarily a Mathieu subspace, where k is a field of characteristic zero, which gives a negative answer to Question 4.1 proposed by van den Essen, Wright and Zhao in [4]. Note that the 2-dimensional Jacobian conjecture is equivalent to saying that, the image of any derivation D of k[x,y] with divergence zero and 1∈ImD is a Mathieu subspace.