Compute the missing binary-family solution at the 2π alignment resonance

Construct the relative periodic orbit of the planar Newtonian three-body problem in the circumbinary configuration with equal-mass bodies B₀ and B₁ and a comparatively small body B₂, based on two consecutive alignments with angle θ = 2π.

Background

The paper studies relative periodic-orbit families in the planar three-body problem by imposing matching conditions at consecutive alignments of the three bodies. In the circumbinary configuration, bodies B₀ and B₁ have equal masses, while B₂ is much less massive and is intended to follow a circumbinary orbit. The continuation calculations exhibit a turning point and approach the resonance θ = 2π, but the continuation does not reach a solution exactly at that resonance.

The unresolved task is to determine whether and how the missing θ = 2π relative periodic orbit can be computed for this binary family. Such a solution would complete the family’s resonance structure and provide an additional coherent three-body trajectory relevant to restricted four-body models with a binary primary.