Consider the ellipsoid $M \subset \mathbb{R}^3$ defined by
$$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1,$$
where $0 < a < b < c$, equipped with the usual Riemannian induced metric from $\mathbb{R}^3$. Let $p \in M$ be a non-umbilic point. I've seen the claim that the set of singularities of the Riemannian exponential map
$$\exp_p\colon T_pM \rightarrow M$$
is an ellipse in this case. Note that I'm talking about conjugate points in the tangent space, not down on the manifold $M$, where this set might be quite ugly. In other words, I'm looking at the set
$$C_p = \{ u \in T_pM : d(\exp_p)(u)~\hbox{is singular} \}.$$
My question is: is there a nice description of $C_p$? Is it an ellipse, at least at most points?
I'm also interested in the case where $b = c$ or $a = b$, as long as it's not a sphere, basically. For the sphere $C_p$ is always a circle, of course.