On the nilpotent commutator of a nilpotent matrix

We study the structure of the nilpotent commutator $\text{nb}$ of a nilpotent matrix $B$. We show that $\text{nb}$ intersects all nilpotent orbits for conjugation if and only if $B$ is a square–zero matrix. We describe nonempty intersections of $\text{nb}$ with nilpotent orbits in the case the $n\times n$ matrix $B$ has rank $n-2$. Moreover, we give some results on the maximal nilpotent orbit that $\text{nb}$ intersects nontrivially.