Doubly symmetric orbits about the collinear Lagrangian points

The forms and stability of three families of periodic solutions in the three-dimensional case of the re- stricted problem are discussed using the two-point boundary value theory and computing the eigenvalues of the Jacobians of the solutions. It is found that stable members of low inclinations in the case of L2 and L8 indeed exist. The question of a possible new integral in the neighborhood of these points is discussed, to- gether with the equivalent case of galactic motions. It is concluded that a new integral with a single-valued gradient cannot exist unless its gradient is collinear to the gradient of the energy integral along all "singular" closed solutions. "Ordinary" solutions, on the other hand, are never found after extensive search by many investigators; and thus the above conclusion seems to be general.