Let $p$ be a point in a Riemannian manifold, and take a normal neighborhood nearby $p$. If a field $X$ on a Riemannian manifold satisfies the condition
$$\mathcal L_X\partial r=[X,\partial r]=0$$
where $\partial r$ is the radial coordinate (which is also $\nabla r$, where $r$ is the distance from $p$) then $X$ is a Jacobi field along all radial geodesics from $p$ (in the normal neighborhood).
(I'm talking about the situation of this post.)
Now the question is: under which conditions does the converse hold? It should be true that the Jacobi equation has more solutions than the given condition (cfr. Petersen, Riemannian geometry). Is this really true? Does a dimension argument work for something like Lie derivative which is not tensorial (Petersen says "the first equation is first order, the Jacobi equation is second order, so the latter has more solutions")?
And, if this is true, is there any known condition which implies the equation with the Lie derivative under the hypotesis "X is Jacobi"?
Sorry for the vague kind of question: the point is that I would like to have the condition in order to prove something, but I can't assume it in general.