Context :

Aims : 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 , known as supersoft sources , are believed to be likely progenitors of Type Ia supernovae .

Methods : From the Chandra archive , we selected normal early-type galaxies with the point source detection sensitivity better than 10 ^ { 37 } \mathrm { erg s ^ { -1 } } in order 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 which represent the young end of the age distribution for elliptical galaxies .
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 .
This choice of the energy band was optimized to detect soft emission from thermonuclear-burning on the surface of an accreting white dwarf .
In computing the L _ { X } we included both unresolved emission and soft resolved sources with the color temperature of kT _ { bb } \leq 200 eV .

Results : We find that the X/K luminosity ratios are in a rather narrow range of ( 1.7 - 3.2 ) \cdot 10 ^ { 27 } \mathrm { erg s ^ { -1 } L _ { K, \odot } } .
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 \sim 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 can not 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 .

Conclusions :