  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 $A=\Ker \delta$ and $M_0=\Ker \delta_M$. 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. 
Locally nilpotent module derivations and the fourteenth problem of   Hilbert