  We report the discovery of three new "near-pristine" Lyman Limit Systems
(LLSs), with metallicities ~1/1000 solar, at redshifts 2.6, 3.8 and 4.0, with a
targeted survey at the Keck Observatory. High resolution echelle spectra of
eight candidates yielded precise column densities of hydrogen and weak, but
clearly detected, metal lines in seven LLSs; we previously reported the one
remaining, apparently metal-free LLS, to have metallicity <1/10000 solar.
Robust photoionisation modelling provides metallicities [Si/H] = -3.05 to
-2.94, with 0.26 dex uncertainties (95% confidence) for three LLSs, and [Si/H]
>~ -2.5 for the remaining four. Previous simulations suggest that near-pristine
LLSs could be the remnants of PopIII supernovae, so comparing their detailed
metal abundances with nucleosynthetic yields from supernovae models is an
important goal. Unfortunately, at most two different metals were detected in
each new system, despite their neutral hydrogen column densities
(10^{19.2-19.4} cm^{-2}) being two orders of magnitude larger than the two
previous, serendipitously discovered near-pristine LLSs. Nevertheless, the
success of this first targeted survey for near-pristine systems demonstrates
the prospect that a much larger, future survey could identify clear
observational signatures of PopIII stars. With a well-understood selection
function, such a survey would also yield the number density of near-pristine
absorbers which, via comparison to future simulations, could reveal the
origin(s) of these rare systems.
