A remark on vanishing cycles with two strata

Suppose that the critical locus $\mathrm{\Sigma }$ of a complex analytic function $f$ on affine space is, itself, a space with an isolated singular point at the origin $\text{0}$, and that the Milnor number of $f$ restricted to normal slices of $\mathrm{\Sigma }-\{\text{0}\}$ is constant. Then, the general theory of perverse sheaves puts severe restrictions on the cohomology of the Milnor fiber of $f$ at $\text{0}$, and even more surprising restrictions on the cohomology of the Milnor fiber of generic hyperplane slices.