3D homogeneous potentials generating two-parametric families of orbits on the outside of a material concentration

We study three-dimensional homogeneous potentials V = V ( x , y , z ) of degree m which are created outside a finite concentration of matter and they produce a preassigned two-parametric family of spatial regular orbits given in the solved form f ( x , y , z ) = c 1 , g ( x , y , z ) = c 2 ( c 1 , c 2 = {\rm const}). These potentials have to satisfy three linear PDEs; two of them come from the Inverse Problem of Newtonian Dynamics and the last one is the well-known ” Laplace’s equation ”. Our aim is to find common solutions for these three PDEs. Besides that we consider that the functions f and g are also homogeneous in the variables x , y , z of any degree and can be represented uniquely by the ” slope functions ” α ( x , y , z ) and β ( x , y , z ) which are homogeneous of zero degree. Then, we impose three differential conditions on the orbital functions ( α , β ). If they are satisfied for a specific value of m , then we can find the potential by quadratures. The values obtained for m so far are consistent with familiar gravitational and electrostatic and quadratic potentials. Finally, pertinent examples are given and cover all the cases.