Some results on homogeneous locally nilpotent $R$-derivations on $R[X,Y,Z]$

Let $k$ be a field of characteristic zero and $R$ a $k$-algebra. In this paper we study homogeneous $R$-lnds $D$ on $R[X,Y,Z]$ with respect to the standard weights $(1,1,1)$. We show that when $R$ is a PID, $rank(D)$ can be at most $2$ if $\mathrm{deg}(D)⩽3$. As a consequence we obtain a certain class of homogeneous lnds on $k^{[4]}$ whose kernel is $k^{[3]}$. Further when $R$ is a Dedekind domain, we give a bound for minimum number of generators of $\ker (D)$ as an $R$-algebra if $\mathrm{deg}(D)⩽3$.