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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 × 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 . [ABSTRACT FROM AUTHOR]

We study the set [image omitted]( [image omitted]) of all possible Jordan canonical forms of nilpotent matrices commuting with a given nilpotent matrix B. We describe [image omitted]( [image omitted]) in the special case when B has only one Jordan block and discuss some consequences. In the general case, we find the maximal possible index of nilpotency in the set of all nilpotent matrices commuting with a given nilpotent matrix. We consider several examples. [ABSTRACT FROM AUTHOR]