  We compute the Bernstein-Sato polynomial of $f$, a function which given a
pair $(M,v)$ in $X = M_n(\mathbf{C}) \times \mathbf{C}^n$ tests whether $v$ is
a cyclic vector for $M$. The proof includes a description of shift operators
corresponding to the Calogero-Moser operator $L_k$ in the rational case.
