three-body problem is to add a third center of force midway between the original two centers, that exerts only a linear Hooke force (confer Hooke's law). The next generalization is to augment the inverse-square force laws with a force that increases linearly with distance. The final set of generalizations is to add two fixed centers of force at positions that are imaginary numbers, with forces that are both linear and inverse-square laws, together with a force parallel to the axis of imaginary centers and varying as the inverse cube of the distance to that axis. The solution to the original Euler problem is an approximate solution for the motion of a particle in the gravitational field of a prolate body, i.e., a sphere that has been elongated in one direction, such as a cigar shape. The corresponding approximate solution for a particle moving in the field of an oblate spheroid (a sphere squashed in one direction) is obtained by making the positions of the two centers of force into imaginary numbers. The oblate spheroid solution is astronomically more important, since most planets, stars and galaxies are approximately oblate spheroids; prolate spheroids are very rare. The analogue of the oblate case in general relativity is a Kerr black hole. The geodesics around this object are known to be integrable, owing to the existence of a fourth constant of motion (in addition to energy, angular momentum, and the magnitude of four-momentum), known as the Carter constant. Euler's oblate three body problem and a Kerr black hole share the same mass moments, and this is most apparent if the metric for the latter is written in Kerr–Schild coordinates. The analogue of the oblate case augmented with a linear Hooke term is a Kerr–de Sitter black hole. As in Hooke's law, the cosmological constant term depends linearly on distance from the origin, and the Kerr–de Sitter spacetime also admits a Carter-type constant quadratic in the momenta. Mathematical solutions Original Euler problem In the original Euler problem, the two centers of force acting on the particle are assumed to be fixed in space; let these centers be located along the x-axis at ±a. The particle is likewise assumed to be confined to a fixed plane containing the two centers of force. The potential energy of the particle in the field of these centers is given by where the proportionality constants μ1 and μ2 may be