Start by considering a large mass, spherical of radius $r$ or point, at the origin. Note that at perihelion or aphelion the relative motion of an orbiting object is tangent to the center of mass.
Any orbit with distance $A > r$ from the origin to aphelion and distance $P$ to perihelion is characterized entirely by a point on in the unit sphere (corresponding, say, to the point of perihelion) along with a point on the unit circle (giving the direction of tangent motion at perihelion).
We can therefore think about the set of $A,P$ orbits in terms of unit vector fields on the unit 2-sphere. More generally, the set of orbits with energy $E$ and perihelion distance $P$ will have the same property.
For a spherical center mass the two are completely equivalent. If the central mass is not spherical, there is still an equal-energy surface $S_E$ of perihelia. $S_E$ will no longer be spherical but will (usually?) be topologically the same as a 2-sphere.
If $S_E$ is topologically spherical, then vector fields on $S_E$ are subject to simple topological analysis, including the famous "hedgehog theorem" stating that there are no non-vanishing continuous vector fields on $S_E$.
I no longer believe it is reasonable to think in terms of perihelion and aphelion, since that assumes an orbit in the first place. We can still think in terms of tangents at some equipotential surface, but have to allow for the full range of kinetic energies (speeds). We are thus characterizing a set of geodesics by a point on the sphere plus a tangent vector, rather than just a point on the unit circle.