  The flow of baryons to and from a galaxy, which is fundamental for galaxy
formation and evolution, can be studied with galaxy-metal absorption system
pairs. Our search for galaxies around CIV absorption systems at $z\sim5.7$
showed an excess of photometric Lyman-$\alpha$ emitter (LAE) candidates in the
fields J1030+0524 and J1137+3549. Here we present spectroscopic follow-up of 33
LAEs in both fields. In the first field, three out of the five LAEs within
10$h^{{-}1}$ projected comoving Mpc from the CIV system are within $\pm500$ km
s$^{{-}1}$ from the absorption at $z_{\text{CIV}}=5.7242\pm0.0001$. The closest
candidate (LAE 103027+052419) is robustly confirmed at
$212.8^{+14}_{-0.4}h^{-1}$ physical kpc from the CIV system. In the second
field, the LAE sample is selected at a lower redshift ($\Delta z\sim0.04$) than
the CIV absorption system as a result of the filter transmission and, thus, do
not trace its environment. The observed properties of LAE 103027+052419
indicate that it is near the high-mass end of the LAE distribution, probably
having a large HI column density and large-scale outflows. Therefore, our
results suggest that the CIV system is likely produced by a star-forming galaxy
which has been injecting metals into the intergalactic medium since $z>6$.
Thus, the CIV system is either produced by LAE 103027+052419, implying that
outflows can enrich larger volumes at $z>6$ than at $z\sim3.5$, or an
undetected dwarf galaxy. In either case, CIV systems like this one trace the
ionized intergalactic medium at the end of cosmic hydrogen reionization and may
trace the sources of the ionizing flux density.
