A remark on vanishing cycles with two strata
  Suppose that the critical locus $\Sigma$ of a complex analytic function $f$ on affine space is, itself, a space with an isolated singular point at the origin $\0$, and that the Milnor number of $f$ restricted to normal slices of $\Sigma-\{\0\}$ is constant. Then, the general theory of perverse sheaves puts severe restrictions on the cohomology of the Milnor fiber of $f$ at $\0$, and even more surprising restrictions on the cohomology of the Milnor fiber of generic hyperplane slices. 
L\^e D\~ung Tr\'ang and David B. Massey
math.AG