  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.
