On locally nilpotent derivations of polynomial algebra in three variables

In this paper we investigate locally nilpotent derivations on the polynomial algebra in three variables over a field of characteristic zero. We introduce an iterating construction giving all locally nilpotent derivations of rank $2$. This construction allows to get examples of non-triangularizable locally nilpotent derivations of rank $2$. We also show that the well-known example of a locally nilpotent derivation of rank $3$, given by Freudenburg, is a member of a large family of new examples of rank $3$ locally nilpotent derivations. Our approach is based on considering all locally nilpotent derivations commuting with a given one. We obtain a characterization of locally nilpotent derivations with a given rank in terms of sets of commuting locally nilpotent derivations.