  We model Pop III star formation in different FUV and X-ray backgrounds,
including radiation feedback from protostars. We confirm previous results that
a moderate X-ray background increases the number of Pop III systems per unit
cosmological volume, but masses and multiplicities of the system are reduced.
The stellar mass function also agrees with previous results, and we confirm the
outward migration of the stars within the protostellar discs. We find that
nearly all Pop III star systems are hierarchical, i.e., binaries of binaries.
Typically, two equal-mass stars form near the centre of the protostellar disc
and migrate outward. Around these stars, mini-discs fragment forming binaries
that also migrate outward. Stars may also form at Lagrange points L4/L5 of the
system. Afterward, star formation becomes more stochastic due to the large
multiplicity, and zero-metallicity low-mass stars can form when rapidly ejected
from the disc. Stars in the disc often have eccentric orbits, leading to a
periodic modulation of their accretion rates and luminosities. At the
pericenter, due to strong accretion, the star can enter a red-supergiant phase
reaching nearly Eddington luminosity in the optical bands ($m_{\rm AB} \sim 34$
for a $100~M_{odot}$ star at $z=6$). During this phase, the star, rather than
its nebular lines, can be observed directly by JWST, if sufficiently magnified
by a gravitational lens. The $\sim 10,000$ AU separations and high
eccentricities of many Pop III star binaries in our simulations are favorable
parameters for IMBH mergers - and gravitational waves emission - through
orbital excitation by field stars.
