On the nilpotent commutator of a nilpotent matrix

We study the structure of the nilpotent commutator N (B) of a nilpotent matrix B. We show that N (B) intersects all nilpotent orbits for conjugation if and only if B is a square-zero matrix. We describe nonempty intersections of N (B) with nilpotent orbits in the case the n x n matrix B has rank n - 2. Moreover, we give some results concerning the inverse image of the map taking B to the maximal nilpotent orbit intersecting N (B).