Blowing up the zero section for "Chasse au Canard" (some new kind of geometric canards)

In this paper "Canard cycles and center manifolds" one encounters the blowing up of a non isolated set or manifold of singularities of a vector field or a singular foliation. This is a generalization of the standard blowup of an isolated singularity. This process inspired me to ask the following question:

Let we have a Riemannian manifold. Then we have the geodesic flow on $$TM$$ whose singular set is the zero section. I wonder what will be happen (and what kind of geometric results will appear) if we simulate the same process of the above linked paper in this case of non isolated singular set, i.e. the zero section of the geodesic flow?