The existence and stability of generalized planar central configurations of the trapezoidal type with a non-spherical body at their centre

The existence of trapezoidal planar central configurations (CCs) without a central body and with a spherical central body (the classic case), and also with a central body in the form of an ellipsoid, either homogeneous or inhomogeneous but consisting of ellipsoidal layers of constant density (generalized cases), is demonstrated. It is shown that such CCs exist in a heliocentric system of coordinates with mutually perpendicular diagonals of the trapezia, at the vertices of which the remaining bodies are positioned. The stability of the generalized trapezoidal planar CCs is investigated. It is established that most versions of such CCs do not possess Lyapunov stability.