  We aim to place upper limits on the combined X-ray emission from the
population of steady nuclear-burning white dwarfs in galaxies. In the framework
of the single-degenerate scenario these systems are believed to be likely
progenitors of Type Ia supernovae. From the Chandra archive, we selected normal
early-type galaxies with the point source detection sensitivity better than
10^37 erg/s to minimize the contribution of unresolved low-mass X-ray binaries.
The galaxies, contaminated by emission from ionized ISM, were identified based
on the analysis of radial surface brightness profiles and energy spectra. The
sample was complemented by the bulge of M31 and the data for the solar
neighborhood. To cover a broad range of ages, we also included NGC3377 and
NGC3585. Our final sample includes eight gas-poor galaxies for which we
determine L_X/L_K ratios in the 0.3-0.7 keV energy band. In computing the L_X
we included both unresolved emission and soft resolved sources with the color
temperature of kT_bb <= 200 eV. We find that the X/K luminosity ratios are in
the range of (1.7-3.2) x 10^27 erg/s/L_K,sun. The data show no obvious trends
with mass, age, or metallicity of the host galaxy, although a weak
anti-correlation with the Galactic NH appears to exist. It is much flatter than
predicted for a blackbody emission spectrum with temperature of ~50-75 eV,
suggesting that sources with such soft spectra contribute significantly less
than a half to the observed X/K ratios. However, the correlation of the X/K
ratios with NH has a significant scatter and in the strict statistical sense
cannot be adequately described by a superposition of a power law and a
blackbody components with reasonable parameters, thus precluding quantitative
constraints on the contribution from soft sources. (abbr.)
