Symmetric trajectories for the 2N-body problem with equal masses

We consider the problem of 2N bodies of equal masses in R-3 for the Newtonian-like weak-force potential r(-sigma), and we prove the existence of a family of collision-free nonplanar and nonhomographic symmetric solutions that are periodic modulo rotations. In addition, the rotation number with respect to the vertical axis ranges in a suitable interval. These solutions have the hip-hop symmetry, a generalization of that introduced in [19], for the case of many bodies and taking account of a topological constraint. The argument exploits the variational structure of the problem, and is based on the minimization of Lagrangian action on a given class of paths.