Periodic orbits in a three-dimensional potential

Two families of periodic orbits in a three-dimensional potential of astronomical interest are described. The potential is the one suggested by Martinet and Magnenat. All orbits have the same energy and start from the (x,y) plane with a velocity perpendicular to this plane. An orbit belonging to family 1P (resp. 2P) crosses once (resp. twice) the plane (x,y) with ż > 0 before closing on itself. Orbits of family 1P are found to be approximately plane. This unexpected property is demonstrated using a theorem by Moser.