Suppose a massive, perfectly spherical ball is fixed on a point in space, and that there is a second free particle of negligible mass. I have noticed that the magnitude of the force exerted on this particle, according to the classical laws of gravitation, is independent of all functions of space, with the exception of radius $r$, in the spherical coordinate system $(r, \theta, \phi)$. In other words, along all points in space where the radial argument $r$ is constant (surfaces of constant $r$), the magnitude of the force exerted on the particle by the gravitating object is constant
It recently got me thinking, suppose instead I have two massive, perfectly spherical balls fixed in space with a constant distance $d$ separating them. Is there a coordinate system $(\psi, \eta, \xi)$, such that along the surface of constant $\psi$ (analogous to the coordinate $r$ in the first example), the force of gravitation exerted on a free particle with negligible mass is constant?
The reason I ask this question is because sometimes a change in the choice of coordinates may simplify equations drastically, and I want to explore and exploit that fact for my own personal studies.
I might be pushing my luck here but if a coordinate system does exist, please also include in the answer not only the name of the system, but also the coordinate transformation as well as the inverse transformation. However, if no such coordinate system exists, suggest a way for me to construct one (if possible).
Now I am not that strong in Maths, I am really just a high school student. This question is related to physics which I am highly interested in but I believe the nature of it falls under the domain of Mathematics.
I appreciate any help, thank you.