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...

We prove that an irreducible polynomial derivation in positive characteristic is a Jacobian derivation if and only if there exists an (n-1)-element p-basis of its ring of constants. In the case of two variables we characterize these derivations in terms of their divergence and some nontrivial constants.