  We study the set $\partition{\nb}$ of all possible Jordan canonical forms of
nilpotent matrices commuting with a given nilpotent matrix $B$. We describe
$\partition{\nb}$ in the special case when $B$ has only one Jordan block. 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.
