  We explore the chemo-dynamical properties of a sample of very metal-poor
(VMP) stars selected from the Hamburg/ESO survey, matched with Gaia EDR3, in
the phase-space identified by the three integrals of motion ($L_z$, $E$,
$I_3$). Disk and halo orbits are separated by using the criteria defined in
Carollo et al. (2021). We found 26 stars with $[Fe/H] \leq -2.5$ possessing
disk kinematics, of which 14 are extremely metal-poor. At these metallicities,
the number of stars with disk kinematics is three times its retrograde
counterpart. In the same range of metallicity we also identified 37 halo stars
most tightly bound to the gravitational potential of the progenitor halo. The
origin of these stars are investigated by comparing the observational results
with simulated galaxies from the Aquarius Project and the IllustrisTNG
simulations. We found two mechanisms of formation of VMP stars with disk
kinematics: accretion from early satellites (which is dominant), and {\it
in-situ} formation. These stars are very old, with ages > 12.5 Gyr ($z$ > 5),
and they are $\alpha$-enriched. Accretion and {\it in-situ} formation are also
found for the retrograde counterparts with being accretion also the dominant
mode. Contributing accreted satellites have stellar masses in the range
$[10^{6}-10^9]$ M_sun, and are very gas-rich. The most bound halo stars are the
oldest detected with a median age of ~ 13.3 Gyr ($z$ ~ 11), and
$\alpha$-enriched. Our finding clearly show that very old, very metal-poor
stars store important information on the first stages of assembly of our Galaxy
and its halo.
