  INTEGRAL/SPI has recently observed a strong and extended emission resulting
from electron-positron annihilation located in the Galactic center region,
consistent with the Galactic bulge geometry, without any counterpart at high
gamma-ray energies, nor in the 1809 keV $^{26}$Al decay line. In order to
explain the rate of positron injection in the Galactic bulge, estimated to more
than 10$^{43}$ s$^{-1}$, the most commonly considered positron injection
sources are type Ia supernovae. However, SN Ia rate estimations show that those
sources fall short to explain the observed positron production rate, raising a
challenging question about the nature of the Galactic positron source. In this
context, a possible source of Galactic positrons could be supernova events of a
new type, as the recently observed SN2003dh/GRB030329, an exploding Wolf-Rayet
star (type Ic supernova) associated with a hypernova/gamma-ray burst; the
question about the rate of this kind of events remains open, but could be
problematically low.
  In this paper, we explore the possibility of positron production and escape
by such an event in the framework of an asymmetric model, in which a huge
amount of $^{56}$Ni is ejected in a cone with a very high velocity; the ejected
material becomes quickly transparent to positrons, which spread out in the
interstellar medium.
