Some previous results obtained by Rasio, Shapiro, & Teukolsky are analyzed within the theoretical framework developed by Bisnovatyi-Kogan et al. The Rasio et al. configurations are shown to belong to a special family of configurations of equilibrium for relativistic clusters with a constant value of the central cutoff energy W0 ~= 114. A new classification of the configurations of equilibrium into three different regimes in the central redshift versus temperature diagram is proposed. The possible confrontation of these results with observational parameters is outlined briefly.