On Maxwell's (n+1)-body problem in the Manev-type field and on the associated restricted problem

The planar symmetrical (n + 1)-body problem in a Manev-type field (featured by a potential of the form alpha/r + beta/r(2)) is being tackled. One prov es that, if n equal masses are initially situated at the vertices of a regu lar polygon centered in the (n + 1)-th mass (Maxwell's model for the rings of Saturn), and if the initial velocities form a vector field symmetrical w ith respect to the central mass, then the polygonal configuration is preser ved all along the motion, but with variable side and with variable rotation around the centre. The motion of every mass relative to the centre is give n by the solution of the Manev-type two-body problem. All possible behaviou rs of the polygonal solution are surveyed, and the equilibria of the proble m are pointed out. One associates a restricted problem to them, for which t he Jacobi integral is proved to exist.