Distribution in integral of motions for the nearby and distant halo sample. Top panel: L_z–E_tot for the stars in our nearby halo sample from Gaia EDR3. We have marked the different structures studied in this paper. Bottom panel: same as the top panel, but now showing the energy and angular momentum for the distant tracers associated with the VOD and the HAC. Estimates for the uncertainty in L_z and E_tot are given as error bars for the distant tracers only (as they are large). We note that the HStr are mostly selected in another subspace, namely L_z − L_⊥, so we display (the region occupied by) its members differently.