  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 $Im D$ of every $k$-derivation $D$ of $k[x, y]$ such that $1\in Im D$ and
$div D=0$ is a Mathieu subspace of $k[x, y]$.
