  We measure the strength of HeII$\lambda$4686 nebular emission in passively
evolving ("retired") galaxies, aiming to constrain their populations of hot
accreting white dwarfs (WDs) in the context of the single degenerate (SD)
scenario of Type Ia supernovae (SNe Ia). In the SD scenario, as a WD burns
hydrogen-rich material accreted from a companion star, it becomes a powerful
source of ionizing UV emission. If significant populations of such sources
exist in galaxies, strong emission in the recombination lines of HeII should be
expected from the interstellar medium. To explore this conjecture, we select
from the Sloan Digital Sky Survey ~11 500 emission line galaxies with stellar
ages >1 Gyr showing no signs of AGN activity and co-add their spectra in bins
of stellar population age. For the first time, we detect HeII$\lambda$4686
nebular emission in retired galaxies and find it to be significantly weaker
than that expected in the SD scenario, especially in the youngest age bin (1-4
Gyr) where the SN Ia rate is the highest. Instead, the strength of the observed
HeII$\lambda$4686 nebular emission is consistent with post-asymptotic giant
branch stars being the sole ionizing source in all age bins. These results
limit populations of accreting WDs with photospheric temperatures ($T_{\rm
eff}$) in the range ~(1.5-6)$\cdot 10^5$ K to the level at which they can
account for no more than ~5-10% of the observed SN Ia rate. Conversely, should
all WD progenitors of SN Ia go through the phase of steady nuclear burning with
$T_{\rm eff}$~(1.5-6)$\cdot 10^5$ K, they do not increase their mass by more
than ~0.03 $M_\odot$ in this regime.
