A note on the geometry of the two-body problem on $S^2$

Leveraging on the results of arXiv:2210.13644 , we carry out an investigation of the algebraic three-fold $\Sigma_{C,h}$, the common level set of the Hamiltonian and the Casimir, for the two-body problem for equal masses on $S^2$ subject to a gravitational potential of cotangent type. We determine the topology of its compactification $\overline{\Sigma}_{C,h}$ and how it bifurcates with respect to the admissible values of $(C,h)$, ($C$ being the fixed value of the Casimir and $h$ the fixed value of the Hamiltonian). This bifurcation diagram is actually equal to the bifurcation diagram that describes relative equilibria. We also prove that for $h$ sufficiently negative $\Sigma_{C,h}$ is equipped with a global contact form obtained from the environment symplectic form via a suitable Liouville vector field.