Mechanical systems are considered, consisting of material points of equal masses located at the vertices of regular polyhedra (Plato's solids) and connected by weightless non-deformable rods. The problem of the influence of higher-order moments of inertia (ie, taking into account the properties of higher-order terms in the expansion of the potential) on the motion of these bodies fixed at the center of mass in the central Newtonian force field is investigated. Stationary motions and equilibrium positions are found and their stability is investigated. A bifurcation diagram is presented on the plane of the constants of the energy and area integrals. An interesting fact is noted: for the bodies under consideration, the dimension of the body element (vertex, edge, face), by which it is drawn to the attracting center in stationary motions and equilibrium positions, coincides with the degree of instability.