Singular holomorphic functions for which all fibre-integrals are smooth

For a germ (X,0) of normal complex space of dimension n + 1 with an isolated singularity at 0 and a germ f: (X,0) → (ℂ,0) of holomorphic function with df(x) ≤ 0 for x ≤ 0, the fibre-integrals     ${\displaystyle s\mapsto {\int }_{f=s}\varrho \omega \prime \bigwedge \overline{\omega \prime \prime },\varrho \in C^{\infty }\_\left\{c\right\}\left(X\right),\omega \prime ,\omega \prime \prime \in {\Omega }_X^n}$, are ${\displaystyle C^{\infty }}$ on ℂ* and have an asymptotic expansion at 0. Even when f is singular, it may happen that all these fibre-integrals are ${\displaystyle C^{\infty }}$. We study such maps and build a family of examples where also fibre-integrals for ${\displaystyle \omega \prime ,\omega \prime \prime \in {⍹}_X}$, the Grothendieck sheaf, are ${\displaystyle C^{\infty }}$.