  Star formation in relic HII regions of the first stars is investigated using
magneto-hydrodynamical simulations with a nested grid method that covers 10
orders of magnitude in spatial scale and 20 orders of magnitude in density
contrast. Due to larger fraction of H_2 and HD molecules, its prestellar
thermal evolution is considerably different from that in the first star
formation. Reflecting the difference, two hydrostatic cores appear in a nested
manner: a protostar is enclosed by a transient hydrostatic core, which appears
during the prestellar collapse. If the initial natal core rotates fast at a
rate with rotational to gravitational energy ratio \beta_0 > 0.01-0.1, the
transient hydrostatic core fragments to \sim 10 M_\odot sub-cores at density
\sim 10^9 cm^-3. With smaller rotation energy, fragmentation occurs at higher
density while a single protostar forms without fragmentation if rotation is
extremely slow with \beta_0 < 10^-6 -10^-5. If magnetic field is present, these
threshold values of \beta_0 is boosted owing to angular momentum transport by
the magnetic breaking. Magnetic field also drives the protostellar outflows.
With strong magnetic field, two distinct outflows are observed: The slower one
emanates from the transient hydrostatic core, while the faster one from the
protostar. These flows may affect the final stellar mass by ejecting some of
masses in the initial core, and also may play some role in driving and
maintenance of interstellar turbulence in young galaxies.
