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We prove for generalisations of quasi-homogeneous $n$-body problems withcenter of mass zero and $n$-body problems in spaces of negative constantGaussian curvature that if the masses and rotation are fixed, there exists, forevery order of the masses, at most one equivalence class of relative equilibriafor which the point masses lie on a circle, as well as that there exists, forevery order of the masses, at most one equivalence class of relative equilibriafor which all but one of the point masses lie on a circle and rotate around theremaining point mass. The method of proof is a generalised version of a proofby J.M. Cors, G.R. Hall and G.E. Roberts on the uniqueness of co-circularcentral configurations for power-law potentials.

Finiteness of polygonal relative equilibria for generalised  quasi-homogeneous $n$-body problems and $n$-body problems in spaces of  constant curvature