Locally nilpotent module derivations and the fourteenth problem of Hilbert

Given a locally nilpotent derivation on an affine algebra $B$ over a field $k$ of characteristic zero, we consider a finitely generated $B$-module $M$ which admits a locally nilpotent module derivation ${\delta }_M$ (see Definition 1.1 below). Let $A=\text{Ker}\delta$ and $M_0=\text{Ker}{\delta }_M$. We ask if $M_0$ is a finitely generated $A$-module. In general, there exist counterexamples which are closely related to the fourteenth problem of Hilbert. We also look for some sufficient conditions for finite generation.