On the nilpotent commutator of a nilpotent matrix

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