cores may fragment into clusters of massive (MS) or very massive stars (VMS). [@Nakamura1999] found $3\,M_{\odot}$ as the mass of the first stars which may grow up to $\sim 16\,M_{\odot}$ by the accretion. The IMF of Population III could thus be bimodal with peaks at $\sim 1-2\,M_{\odot}$ and $\sim
10^{2}\,M_{\odot}$ [@Nakamura2001]. [@Tumlinson2004] defined *strong VMS hypothesis* (“The first generation were exclusively VMS") and the *weak VMS hypothesis* (“The first generation included VMS in addition to MS with $M\lesssim
50\,M_{\odot}$").

The structure, evolution and properties of the first stars have been modelled in several studies including the PaperI of this series [@Bahena2010]. The results showed that the Population III stars are smaller but more luminous and hotter than the normal metal-richer Population I an II stars of equal masses.

The massive luminous normal stars have a high mass loss mainly due to the stellar wind driven by radiative pressure in lines of metals [@Castor1975 the so called CAK-wind]. This process cannot take place in the metal-free stars [@Bromm2003]. This is why practically all models of first-stars evolution have neglected completely the mass loss.

Several recent studies, however, revealed that even in the absence of CAK-wind there may occur a considerable mass 