Dziobek's configurations in restricted problems and bifurcation (Q1766273)

The author considers central configurations in the restricted \((n+1)\)-body problem and investigates symmetry properties of the restricted \((n+1)\)-body problem in the \((n-1)\)-dimensional space. In particular, the author analyzes five bodies in a three-dimensional configuration with four particles of equal masses \(m\) and a fifth unitary mass located in their convex hull. The following theorem is proved: if \( 0<m<1/16\), then every central configuration of this problem has a plane of symmetry. Some particular cases are discussed.