Images of locally nilpotent derivations acting on ideals of polynomial algebras

Let $k$ be a field of characteristic zero, and $k^{[n]}:=k[x_1,x_2,\ldots,x_n]$ the polynomial algebra in $n$ variables over $k$. The LND Conjecture asserts that the image of a locally nilpotent derivation of $k^{[n]}$ acting on an ideal of $k^{[n]}$ is a Mathieu—Zhao subspace. This conjecture is still open for any $n\geq 2$, which arose from the Jacobian Conjecture. In this paper, we show that the LND Conjecture holds in dimension $n=2$ for principal ideals and some other classes of ideals.