  We have searched for ultraluminous X-ray sources (ULXs) in a sample of 28
elliptical and S0 galaxies observed with Chandra. We find that the number of
X-ray sources detected at a flux level that would correspond to a 0.3-10 keV
X-ray luminosity of ~2 x 10^39 ergs/s or greater (for which we have used the
designation very ultraluminous X-ray sources; VULXs) at the distance of each
galaxy is equal to the number of expected foreground/background objects. In
addition, the VULXs are uniformly distributed over the Chandra field of view
rather than distributed like the optical light of the galaxies, strengthening
the argument that the high flux sources are unassociated with the galaxies. We
have also taken the VULX candidate list of Colbert and Ptak and determined the
spatial distribution of VULXs in early-type galaxies and late-type galaxies
separately. While the spiral galaxy VULXs are clearly concentrated toward the
centers of the galaxies, the early-type galaxy VULXs are distributed randomly
over the ROSAT HRI field of view, again indicating that they are not associated
with the galaxies themselves. We conclude that with the exception of two rare
high luminosity objects within globular clusters of the elliptical galaxy
NGC1399, VULXs are generally not found within old stellar systems. However, we
do find a significant population of sources with luminosities of 1-2 x 10^39
ergs/s that reside within the sample galaxies that can be explained by
accretion onto 10-20 solar mass black holes. Given our results, we propose that
ULXs be defined as X-ray sources with L_X(0.3-10 keV) > 2 x 10^39 ergs/s.
