We consider the intersection of the conjugacy class of a nilpotent matrix with the space of upper triangular matrices. We give necessary and sufficient conditions for this intersection to be a union of finitely many orbits for the action by conjugation of the group of invertible upper triangular matrices.
"Upper triangular parts of conjugacy classes of nilpotent matrices with finite number of $B$-orbits." J. Math. Soc. Japan 65 (3) 967 - 992, July, 2013. https://doi.org/10.2969/jmsj/06530967