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) = c1\documentclass[12pt]{minimal}
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\begin{document}$$c_{1}$$\end{document}, g(x, y, z) = c2\documentclass[12pt]{minimal}
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\begin{document}$$c_{2}$$\end{document} (c1\documentclass[12pt]{minimal}
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\begin{document}$$c_{1}$$\end{document}, c2\documentclass[12pt]{minimal}
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\begin{document}$$\; c_{2}$$\end{document} = {\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\documentclass[12pt]{minimal}
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\begin{document}$$x,\; y,\;z$$\end{document} of any degree and can be represented uniquely by the ”slope functions” α(x,y,z)\documentclass[12pt]{minimal}
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\begin{document}$$\alpha (x,y,z)$$\end{document} and β(x,y,z)\documentclass[12pt]{minimal}
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\begin{document}$$\beta (x,y,z)$$\end{document} which are homogeneous of zero degree. Then, we impose three differential conditions on the orbital functions (α,β\documentclass[12pt]{minimal}
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\begin{document}$$\alpha , \; \beta $$\end{document}). 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.
