Numerical search for periodic solutions in the vicinity of the figure-eight orbit: slaloming around singularities on the shape sphere

We present the results of a numerical search for periodic orbits with zero angular momentum in the Newtonian planar three-body problem with equal masses focused on a (narrow) search window bracketing the figure-eight initial conditions. We found eleven solutions that can be described as some power of the “figure-eight” solution in the sense of the topological classification method. One of these solutions, with the seventh power of the “figure-eight”, is a choreography. We show numerical evidence of its stability.