The nature of the Milky Way's stellar halo revealed by the three
  integrals of motion
  We developed a new selection method of halo stars in the phase-space
distribution defined by the three integrals of motion in an axisymmetric
Galactic potential, ($E$, $L_z$, $I_3$), where $I_3$ is the third integral of
motion. The method is used to explore the general chemo-dynamical structure of
the halo based on stellar samples from SDSS-SEGUE DR7 and DR16-APOGEE, matched
with Gaia-DR2. We found, (a) halo stars can be separated from disk stars by
selecting over (1) $0 < L_z < 1500$ \kpckms, $(2I_3)^{1/2} > 1000$ \kpckms
(orbital angle $\theta_{\rm orb}$ $>$ 15-20 deg), and $E < -1.5 \times 10^5$
km$^2$ s$^{-2}$, and (2) $L_z < 0$ \kpckms. These selection criteria are free
from kinematical biases introduced by the simple high-velocity cuts adopted in
recent literature; (b) the averaged, or {\it coarse-grained}, halo phased-space
distribution shows a monotonic exponential decrease with increasing $E$ and
$I_3$ like the Michie-Bodenheimer models; (c) the inner stellar halo described
in \citet{carollo2007,carollo2010} is found to comprise a combination of Gaia
Enceladus debris (GE; \citealt{helmi2018}), lowest-$E$ stars (likely in-situ
stars), as well as metal-poor prograde stars missed by the high velocity cuts
selection; (d) the very metal poor outer halo, ([Fe/H] $< -$2.2), exhibits both
retrograde and prograde rotation, with an asymmetric $L_z$ distribution towards
high retrograde motions, and larger $\theta_{\rm orb}$ than those possessed by
the GE dominated inner halo; (e) the Sgr dSph galaxy could induce a long-range
dynamical effect on local halo stars. Implication for the formation of the
stellar halo are also discussed.
