by internal mixing of matter processed by nuclear reactions. One significant observational evidence, which has yet to be explained, is that the number of carbon-enhanced stars increases considerably when decreasing the metallicity below ${\rm [Fe/H]}\approx-2.5$ [@RBS99]. Several of the extremely metal-deficient carbon stars show a large carbon abundance as well as nitrogen enrichment (\[C/Fe\] resp. ${\rm [N/Fe]}\ga2$). In some of these carbon-enriched stars the abundances of $s-$process elements are enhanced, too (see @H00 and reference therein). This obviously implies that in extremely metal-poor stars mechanisms are at work which increase the probability for producing carbon-rich objects.

In the last decades, several theoretical studies of the structural and evolutionary properties of extremely metal-poor and metal-free stars (see for instance the reviews by @VC and @CC and references therein) have shown how the lack of heavy elements — mainly of CNO-cycle catalysts — produces evolutionary features, which are rather peculiar in comparison to stars of finite initial metallicity. One of these properties – investigated by Fujimoto and coworkers [@FIH90; @HIF90; @FII00] – is the evolution during the He-flash. Like their more metal-rich counterparts, metal-free stars with a mass of about $1.0\,M_\odot$ reach the conditions necessary for igniting triple-alpha reactions inside the He 