A generalized Sitnikov problem

In this paper we address a \(n+1\)-body gravitational problem governed by the Newton's laws, where \(n\) primary bodies orbit on a plane \(\Pi\) and an additional massless particle moves on the perpendicular line to \(\Pi\) passing through the center of mass of the primary bodies. We find a condition for that the configuration described be possible. In the case that the primaries are in a rigid motion we classify all the motions of the massless particle. We study the situation when the massless particle has a periodic motion with the same minimal period than primary bodies. We show that this fact is related with the existence of certain pyramidal central configuration.