  (abridged) 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. 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. 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.
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. We show that rotation is a key ingredient to
explain the abundance patterns of CRUMPS stars.
