  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 $2$.
