Images of locally nilpotent derivations of bivariate polynomial algebras over a domain

We study the LND conjecture concerning the images of locally nilpotent derivations, which arose from the Jacobian conjecture. Let R be a domain containing a field of characteristic zero. We prove that, when R is a one-dimensional unique factorization domain, the image of any locally nilpotent R-derivation of the bivariate polynomial algebra R[x, y] is a Mathieu-Zhao subspace. Moreover, we prove that, when R is a Dedekind domain, the image of a locally nilpotent R-derivation of R[x, y] with some additional conditions is a Mathieu-Zhao subspace.