Continuous Families of Central Configurations for Even-Power Homogeneous Potentials

We study central configurations for $n$ bodies interacting through pair potentials \[ U_σ = \fracκσ \sum_{i<j}m_i m_j r_{ij}^σ, \] with logarithmic limit at $σ=0$. For the Newtonian exponent $σ=-1$, the classical Chazy--Wintner--Smale finiteness problem asks whether, for prescribed positive