I know two papers in the mathematical physics literature which might be relevant:

One paper on the two-dimensional Coulomb problem (i.e., with logarithmic potential) was looked at in a paper by Kogan, Perelomov and Semenoff Charge distribution in 2-D electrostatics: A Shell model, about which I heard in a seminar in Bonn in 1992 if memory serves me. I thought the result quite striking and was left with the impression that the resulting configurations ought to be explained by representation theory of some algebraic structure.

There is a also a paper on three-dimensional central potentials by Battye, Gibbons and Sutcliffe Central configurations in three dimensions.

I know two papers in the mathematical physics literature which might be relevant:

One paper on the two-dimensional Coulomb problem (i.e., with logarithmic potential and a charge at infinity) is by Kogan, Perelomov and Semenoff, titled Charge distribution in 2-D electrostatics: A Shell model, about which I first heard in a seminar in Bonn in 1992 if memory serves me. I thought the result quite striking and was left with the impression that the resulting configurations ought to be explained by representation theory of some algebraic structure.

There is a also a paper on three-dimensional central potentials by Battye, Gibbons and Sutcliffe Central configurations in three dimensions.