Are Nonsymmetric Balanced Configurations of Four Equal Masses Virtual or Real?

Balanced configurations of $N$ point masses are the configurations which, in a Euclidean space of high enough dimension, i.e., up to $2(N - 1)$, admit a relative equilibrium motion under the Newtonian (or similar) attraction. Central configurations are balanced and it has been proved by Alain Albouy that central configurations of four equal masses necessarily possess a symmetry axis, from which followed a proof that the number of such configurations up to similarity is finite and explicitly describable. It is known that balanced configurations of three equal masses are exactly the isosceles triangles, but it is not known whether balanced configurations of four equal masses must have some symmetry. As balanced configurations come in families, it makes sense to look for possible branches of nonsymmetric balanced configurations bifurcating from the subset of symmetric ones. In the simpler case of a logarithmic potential, the subset of symmetric balanced configurations of four equal masses is easy to describe as well as the bifurcation locus, but there is a grain of salt: expressed in terms of the squared mutual distances, this locus lies almost completely outside the set of true configurations (i. e., generalizations of triangular inequalities are not satisfied) and hence could lead most of the time only to the bifurcation of a branch of virtual nonsymmetric balanced configurations. Nevertheless, a tiny piece of the bifurcation locus lies within the subset of real balanced configurations symmetric with respect to a line and hence has a chance to lead to the bifurcation of real nonsymmetric balanced configurations. This raises the question of the title, a question which, thanks to the explicit description given here, should be solvable by computer experts even in the Newtonian case. Another interesting question is about the possibility for a bifurcating branch of virtual nonsymmetric balanced configurations to come back to the domain of true configurations.