The equations of motion of the three-dimensional restricted three-body problem with oblateness are found to allow the existence of out-of-plane equilibrium points. These points lie in the (x − z) plane almost directly above and below the center of each oblate primary. Their positions can be determined numerically and are approximated by series expansions. The effects of their existence on the topology of the zero-velocity curves are considered and their stability is explored numerically.

We discuss the existence, location, and stability of the collinear equilibrium points of a generalized Hill problem with radiation of the primary (the Sun) and oblateness of the secondary (the planet), and present some remarkable fractals created as basins of attraction of Newton's method applied for their computation in several cases of the parameters.

The existence of new equilibrium points is established in the restricted three-body problem with equal prolate primaries. These are located on the Z-axis above and below the inner Eulerian equilibrium point L 1 and give rise to a new type of straight-line periodic oscillations, different from the well known Sitnikov motions. Using the stability properties of these oscillations, bifurcation points are found at which new types of families of 3D periodic orbits branch out of the Z-axis consisting of orbits located entirely above or below the orbital plane of the primaries. Several of the bifurcating families are continued numerically and typical member orbits are illustrated.

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