A note on homogeneous rank $2$ locally nilpotent derivations on $k[X,Y,Z]$

In this article we show that for every prime number $p$, any irreducible homogeneous locally nilpotent derivations of rank $2$ and degree $p-2$ are triangularizable. Further, we describe the structure of irreducible non-triangularizable homogeneous locally nilpotent derivations of rank $2$ and degree $pq-2$, where $p,q$ are prime numbers. Consequently, we give explicit descriptions of the generators of the image ideals of certain homogeneous locally nilpotent derivations of rank $2$.