  Helium accretion induced explosions in CO white dwarfs (WDs) are considered
promising candidates for a number of observed types of stellar transients,
including supernovae (SNe) of Type Ia and Type Iax. However, a clear favorite
outcome has not yet emerged. We explore the conditions of helium ignition in
the white dwarf and the final fates of helium star-WD binaries as function of
their initial orbital periods and component masses. We compute 274 model binary
systems with the Binary Evolution Code (BEC), where both components are fully
resolved. Stellar and orbital evolution is computed simultaneously, including
mass and angular momentum transfer, tides, and gravitational wave emission, as
well as differential rotation and internal hydrodynamic and magnetic angular
momentum transport. We find that helium detonations are expected only in
systems with the shortest initial orbital periods, and for initially massive
white dwarfs (MWD > 1.0 MSun ) and lower mass donors (Mdonor < 0.8 MSun), with
accumulated helium layers mostly exceeding 0.1 MSun. Upon detonation, these
systems would release the donor as a hypervelocity pre-WD runaway star, for
which we predict the expected range of kinematic and stellar properties.
Systems with more massive donors or initial periods exceeding 1.5 h will likely
undergo helium deflagrations after accumulating 0.1 - 0.001 MSun of helium.
Helium ignition in the white dwarf is avoided in systems with helium donor
stars below - 0.6 MSun, and lead to three distinctly different groups of double
white dwarf systems. The size of the parameter space open to helium detonation
corresponds to only about 3 % of the galactic SN Ia rate, and to 10 % of the SN
Iax rate, while the predicted large amounts of helium (>0.1 MSun) in
progenitors cannot easily be reconciled with observations of archetypical SN
Ia. ...
