An infinite fiber of shifted singular-value data

There is a nonconstant smooth one-parameter family of complex $9\times9$ matrices whose members have the same singular values for every complex scalar shift, while no two distinct members are unitarily similar; each member is nilpotent with Jordan type $(3,3,3)$. Consequently no uniform finite bound on the number of unitary similarity classes with prescribed shifted singular values exists in dimension nine. The family consists of block matrices with superdiagonal blocks $P^{1/2}$ and $Q^{1/2}$ for positive definite Hermitian $3\times3$ matrices $P,Q$. An exact block determinant identity shows that the polynomial $\det[uI+s(P+Q)+PQ]$ determines all shifted singular values, so these data are fixed by nine real coefficients. A ten-parameter family with no repeated unitary class and the constant-rank theorem then yield a positive-dimensional fiber.