  The Li-rich stars are a class of rare objects with A(Li) higher than that of
other stars in the same evolutionary stage. Their origin is still debated and
valuable routes are the Cameron-Fowler mechanism, mass transfer process in a
binary system or engulfment of small bodies. Metal-poor ([Fe/H]<-1 dex) stars
are only a small fraction of the entire population of Li-rich stars. We
observed with MIKE at the Magellan Telescope the metal-poor ([Fe/H]=-3.95+-0.11
dex) giant star HE0057-5959, deriving A(Li)=+2.09+-0.07 dex. Such A(Li) is
higher by about 1 dex than that of other stars in the same evolutionary stage.
A previous analysis of the same target suggested that its high A(Li) reflects a
still ongoing First Dredge-Up process. We revise the nature of HE0057-5959 by
comparing its stellar parameters and A(Li) with appropriate stellar evolution
models. This comparison rules out that HE0057-5959 is caught during its First
Dredge-Up, being this latter already ended according to the parameters of this
star. Its A(Li), remarkably higher than the typical lithium plateau drawn by
similar giant stars, demonstrates that HE0057-5959 joins the class of the rare
metal-poor Li-rich stars. HE0057-5959 is the most metal-poor Li-rich star
discovered so far. We consider different scenarios to explain this star. No
internal mixing able to activate the Cameron-Fowler mechanism is known for
metal-poor stars at this evolutionary stage. Also the engulfment of planets is
disfavored because such metal-poor stars should not host planets. Finally,
HE0057-5959 is one of the most Na-rich among the Li-rich stars and we found
that a strong excess of Na is common to the three Li-rich stars with [Fe/H]<-3
dex. This finding could support the scenario of mass transfer from a massive
companion star in a binary system, even if we found no evidence of radial
velocity variations.
