  This study explores the metal enrichment signatures attributed to the first
generation of stars (PopIII) in the Universe, focusing on the E-XQR-30 sample.
We aim to identify traces of Pop III metal enrichment by analyzing neutral gas
in the interstellar medium of primordial galaxies and their satellite clumps,
detected in absorption. To chase the chemical signature of PopIII stars, we
studied metal absorption systems in the E-XQR-30 sample, selected through the
detection of the OI absorption line at 1302A. The OI line is a reliable tracer
of HI and allowed us to overcome the challenges posed by the Lyman-$\alpha$
forest's increasing saturation at redshifts above $\sim5$ to identify Damped
Lyman-$\alpha$ systems (DLA). We detected and analyzed 29 OI systems at
$z\geq5.4$, differentiating between proximate DLAs (PDLA) and intervening DLAs.
Voigt function fits were applied to obtain ionic column densities, and relative
chemical abundances were determined for 28 systems. These were then compared
with the predictions of theoretical models. Our findings expand the study of OI
systems at $z\geq5.4$ fourfold. No systematic differences were observed in the
average chemical abundances between PDLAs and intervening DLAs. The chemical
abundances in our sample align with literature systems at $z>4.5$, suggesting a
similar enrichment pattern for this class of absorption systems. A comparison
between these DLA-analogues at $4.5<z<6.5$ with a sample of very metal-poor
DLAs at $2<z<4.5$ shows in general similar average values for the relative
abundances, with the exception of [C/O], [Si/Fe] and [Si/O] which are
significantly larger for the high-$z$ sample. Furthermore, the dispersion of
the measurements significantly increases in the high-redshift bin. This
increase is predicted by the theoretical models and indicates a potential
retention of PopIII signatures in the probed gas. (Abridged)
