Investigating a Special Class of Central Configurations for the Spatial Six Body Problem

The N-body problem in classical mechanics has been researched since a variation of it appeared in Newton's Principia Mathematica.  Given the mutual gravitational attraction between N masses and their initial positions/velocities at current time t₀, can one determine/predict the position of the N masses for all past/future time t ≠ t₀?  It was later shown by Poincaré that the N-body problem is not integrable, so in general one cannot find explicit solutions.  A particular class of explicit solutions for the N-body problem is constituted by central configurations, an ensemble of N bodies in which the acceleration of each mass is proportional to its position vector.  This proportionality constant must be the same for all masses.  We investigate a class of three dimensional central configurations of six bodies in which three bodies of equal mass lie at the vertices of an equilateral triangle, and three other bodies lie on a line perpendicular to the plane of the triangle and passing through its center of mass.  Applying the Laura/Andoyer/Dziobek equations to this configuration transforms the existence question of central configurations into an analytical problem involving a system of equations whose solutions are the masses. We seek to understand when such solutions exist.  Central configurations have applications in astronomy and astrodynamics. The system formed by the Sun, Jupiter and Trojan asteroids is a well known example of a Lagrangian central configuration. Central configurations have been used to model the stability of particles surrounding Saturn, and of formation flights of artificial satellites.