Determinantal representations of closed orbits

In this paper, we show that the orbit of a point mass under a central force f(r)=-αr-2-βr-3 is realized as the hyperbolic curve FA(1,x,y)=0 associated with a nilpotent matrix A. On the contrary, we show that the orbit of motion of particles of infinitesimal mass in the gravitational field described by Schwarzschild geodesic metric is transcendental. In this case, the transcendental orbit has no determinantal representations.