The planar inverse problem for autonomous systems.

The authors study the general version of the inverse problem for planar trajectories and for autonomous dynamical systems possessing three integrals, i.e., for a given three-parametric family of curves f(x,y,a,b) = c they find the potential V(x,y) for which these curves are orbits of a unit mass. All possible cases, depending on the preassigned function f, are classified and in each case the necessary and sufficient conditions for the existence of a solution are established. Among the examples there is the case of the Keplerian conic sections which is studied in detail.