It is well known that the gravitational forces due to a spherical shell of uniform density cancels in the interior of the shell (in three dimensions). Another way to state this is that the gravitational potential is uniform in the interior of the sphere.

Suppose you are given a sphere of fixed radius and you can design a radial force (and associated radial potential) . Can you

(1) Give example of other potentials that are constant in the interior of the sphere?

(2) Characterize all potentials that are uniform in the sphere's interior?

Literature references are particularly appreciated.

I know that the solutions to (2) are a vector space and are a solution to an integral equation, but it is not clear that this is a fruitful line of attack.