A Counterexample to a Generalized Saari's Conjecture with a Continuum of Central Configurations

In this paper we show that in the $n$-body problem with harmonic potential one can find a continuum of central configurations for $n=3$. Moreover we show a counterexample to an interpretation of Jerry Marsden Generalized Saari's conjecture. This will help to refine our understanding and formulation of the Generalized Saari's conjecture, and in turn it might provide insight in how to solve the classical Saari's conjecture for $n\geq 4$.