In this paper we show that the image of any locally finite k-derivation of the polynomial algebra k[x,y] in two variables over a field k of characteristic zero is a Mathieu subspace. We also show that the two-dimensional Jacobian conjecture is equivalent to the statement that the image of every k-derivation D of k[x,y] such that and is a Mathieu subspace of k[x,y].
van den Essen, Arno; Wright, David; and Zhao, Wenhua, "Images of locally finite derivations of polynomial algebras in two variables" (2011). Mathematics Faculty Publications. 1.