  Recent theory predicts that a first star is born with a massive initial mass
of $\gtrsim$ 100 $M_\odot$. Pair instability supernova (PISN) is a common fate
for such a massive star. Our final goal is to prove the existence of PISN and
thus the high mass nature of the initial mass function in the early universe by
conducting {\it abundance profiling}, in which properties of a hypothetical
first star is constrained by metal-poor star abundances. In order to determine
reliable and useful abundances, we investigate the PISN nucleosynthesis taking
both rotating and non-rotating progenitors for the first time. We show that the
initial and CO core mass ranges for PISNe depend on the envelope structures:
non-magnetic rotating models developing inflated envelopes have a lower-shifted
CO mass range of $\sim$ 70--125 $M_\odot$, while non-rotating and magnetic
rotating models with deflated envelopes have a range of $\sim$ 80--135
$M_\odot$. However, we find no significant difference in explosive yields from
rotating and non-rotating progenitors, except for large nitrogen production in
non-magnetic rotating models. Furthermore, we conduct the first systematic
comparison between theoretical yields and a large sample of metal-poor star
abundances. We find that the predicted low [Na/Mg] $\sim$ $-1.5$ and high
[Ca/Mg] $\sim$ $0.5$--$1.3$ abundance ratios are the most important to
discriminate PISN signatures from normal metal-poor star abundances, and
confirm that no currently observed metal-poor star matches with the PISN
abundance. Extensive discussion on the non-detection is finally made.
