On finite generation of kernels of locally nilpotent R-derivations of R[X,Y,Z]

Let R be a noetherian domain containing the field of rationals. We show that if R is Dedekind then the kernel of any locally nilpotent R-derivation of R[X,Y,Z] is a finitely generated R-algebra. Conversely, we show that if R is neither a field nor a Dedekind domain then there exists a locally nilpotent R-derivation of R[X,Y,Z] whose kernel is not finitely generated over R.