Context : The three most iron poor stars presently known ( [ Fe/H ] equal to -5.96 , -5.4 and -4.75 ) are carbon-rich , they are called C-Rich Ultra-Metal Poor Stars ( CRUMPS ) .
The origin of their peculiar surface abundances is not understood .

Aims : We propose a synthetic view of the different models so far proposed to explain the peculiar abundances observed at the surface of the CRUMP stars .
We deduce some expected trends based on nucleosynthetic arguments and look for signatures allowing to discriminate among models .
We discuss the conditions for having CRUMP stars which are He-rich , i.e .
with a mass fraction of helium greater than 0.30 and up to 0.60 .

Methods : We discuss the chemical composition of stars made of interstellar medium mixed with wind material of very metal poor massive stars , with wind plus supernova ejecta and with material extracted from the envelope of early AGB stars .
Rotating and non-rotating models are considered .

Results : The high nitrogen abundances observed in CRUMP stars imply that the material which is responsible for their peculiar abundance pattern must be heavily enriched in primary nitrogen .
We show that rotating stars ( both massive and intermediate mass stars ) can produce the required amount of primary nitrogen , and can also account for the observed enhancements in C , O , Na , Mg and Al .
CRUMP stars formed from wind material of massive stars mixed with small amounts of pristine interstellar medium are He-rich ( helium mass fraction between 0.30 and 0.60 ) , Li-depleted and present low ^ { 12 } C/ ^ { 13 } C ratios ( inferior to 10 in number ) .
Such He-rich stars , if discovered , would confirm that the most metal poor CRUMPs formed from essentially pure wind/envelope material .
They would provide the most direct way to probe the nucleosynthetic outputs of the first generations of stars .

Conclusions : We show that rotation is a key ingredient to explain the abundance patterns of CRUMPS stars and probably also of at least some Carbon-Enhanced Metal Poor ( CEMP ) stars , in particuliar the CEMP-no stars .
Similar non-rotating models , without any extra-mixing , do not succeed to explain the enhancements in the three CNO elements .