to the strong fluctuations in the positions of this part. This fluctuations result in smearing the central peak in both core-halo and low energy uniform states. Considering the center of mass system of reference does not resolve this difficulty, as, despite being dense, the core typically contains only 10 – 20% of the total system mass (see below) and the positions of the core and the center of mass of the system do not usually coincide.

To control the quality of the simulation, we monitored the total energy $\e$ and the total angular momentum $L$. We selected timestep $dt$ small enough to keep the total energy variation within 0.01% of its initial value, usually we used $dt =10^{-5}$, or in rescaled units, $d\t \sim 10^{-4}$. For such time steps, the relative deviation of the angular momentum was within $10^{-14}$.

All the measurements below are presented in the rescaled dimensionless units as defined in Eq. (\[def\]).

uniform and core-halo equilibrium states: comparison to the MF
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To check our simulation procedure and possibly resolve the apparent contradiction between the MF and the particle simulation results [@pet], we first considered the system in what we expected to be a stable or a 