Let $(V,0)\subset (\mathbb{C}^n,0)$, $n\geq 3$ a (germ of) hypersurface given by $\{f=0\}$, $f$ holomorphic. A (germ of) holomorphic vector field $X$ leaves $V$ invariant if ${\rm d}f(X)$ belongs to the ideal generated by $f$.
My question is: For any such $V$, does there exist a vector field tangent to $V$ with an isolated singularity?
When $V$ is smooth or $V$ has an isolated singularity (at the origin) then one can readily exhibit such vector fields. However, for germs of non-isolated singularities I could neither prove this nor find a counterexample.