Branch bifurcation from the degenerate tetrahedral circular orientation

Determine whether a branch of periodic solutions of the tetrahedrally symmetric (n+1)-body problem emanates from the circular orbit whose orbit-plane normal is the body diagonal (1,1,1)/√3, despite the singularity of the reduced Hessian at that orientation.

Background

The computer-assisted search identifies a tetrahedral circular orientation that is a genuine critical point of the orientation equations. However, the great circle has an additional three-fold symmetry, causing the eccentricity block of the reduced Hessian to vanish identically and making the full four-dimensional Hessian singular. Consequently, the nondegenerate continuation theorem used elsewhere in the paper does not apply, and the existence of a bifurcating branch remains unresolved.

References

The orientation is nonetheless a genuine critical point of the orientation equations, forced by the three-fold symmetry. The degeneracy is of the reduced function and not of the problem, and deciding whether a branch emanates from it requires an analysis beyond the scope of this paper.

— Platonic constellations of periodic motions in the $(n + 1)$-body problem  (2609.10242 - Constantineau et al., 9 Sep 2026) in Remark, Section 4, following Theorem 4.2 [label rendered as Remark rem:degenerate]