I confess to be not an expert of analytic geometry, but I have come across the following problem, for which I need an help from experts in this specific field.


I was wondering myself if it is possible to find a real analytic set $X$ in $\mathbb R^n$, of codimension $1$, which comes equipped with a foliation into analytic compact manifolds of codimension $2$, one of whose leaves is exactly the singular locus of $X$.
Or otherwise, why is such a configuration irrealizable?


If the answer could depend on $n$, then I have meet this question working with $n=4$.



Can an analytic set admit such a foliation?
Let $X$ be the cone $x^2+y^2=(z-1)^3$, foliated by circles $z$ constant. Or just take any curve in the plane and foliate by points.


