Inverse problem with two parametric families of planar orbits

As a possible extension of recent work, the following version of the inverse problem in dynamics is studied. Given a two-parametric family f(x,y,b) = c of plane curves, find an autonomous dynamical system for which these curves are orbits. A new linear partial differential equation of the first order is derived for the force components X(x,y) and Y(x,y) corresponding to the given family. With the aid of this equation it is found that, depending on the given function f, the problem may or may not have a solution. Based on given criteria, a full classification is presented of the various cases which may arise.