Common energy surface for weak-force hip-hop solutions

Determine whether the hip-hop symmetric solutions obtained for weak-force potentials lie on a common energy surface.

Background

The paper discusses hip-hop symmetric solutions for the equal-mass $2n$-body problem and their extension to the $(1+2n)$-body problem. Earlier work established families of hip-hop symmetric solutions for weak-force potentials using variational methods, but the energy relationship among those solutions was not resolved. The authors explicitly contrast this unresolved issue with their own Newtonian result, which proves the existence of infinitely many hip-hop symmetric orbits on every compact regular energy surface.