Two-parametric families of orbits produced by 3D potentials inside a material concentration: an application to galaxy models

We study two-parametric families of spatial orbits given in the analytic form \(f(x,y,z)=c_{1}\), \(g(x,y,z)=c_{2}\) (\(c_{1}\), \(c_{2}\) = const.) which are produced by three-dimensional potentials \(V=V(x,y,z)\) inside a material concentration. These potentials must verify two linear partial differential equations (PDEs) which are the basic equations of the 3D Inverse Problem of Newtonian Dynamics and the well-known Poisson’s equation. A suitable class of potentials for this case is the axisymmetric potentials \(V=\mathcal{B}(x^{2}+y^{2}, z)\) which have applications in astrophysical problems. For the given density function \(\rho =\rho (x, y, z)\), \(\rho =\rho _{0}=const\)., or, \(\rho =\rho (z)\) and a pre-assigned family of orbits, three-dimensional potentials producing this family of orbits are found in each case. We focus our interest on the cored, logarithmic potentials and another one of fourth degree describing elliptical galaxies. The two-parametric families of straight lines in 3D space are also considered.