Three-dimensional periodic solutions around equilibrium points in Hill's problem

The three-dimensional periodic solutions originating at the equilibrium points of Hill's limiting case of the Restricted Three Body Problem, are studied. Fourth-order parametric expansions by the Lindstedt-Poincaré method are constructed for them. The two equilibrium points of the problem give rise to two exactly symmetrical families of three-dimensional periodic solutions. The family_HL _2v ^e originating at L₂ is continued numerically and is found to extend to infinity. The family originating at L₁ behaves in exactly the same way and is not presented. All orbits of the two families are unstable.