  Very old isolated neutron stars and white dwarfs have been suggested to be
probes of dark matter. To play such a role, two requests should be fulfilled,
i.e., the annihilation luminosity of the captured dark matter particles is
above the thermal emission of the cooling compact objects (request-I) and also
dominate over the energy output due to the accretion of normal matter onto the
compact objects (request-II). Request-I calls for very dense dark matter medium
and the critical density sensitively depends on the residual surface
temperature of the very old compact objects. The accretion of
interstellar/intracluster medium onto the compact objects is governed by the
physical properties of the medium and by the magnetization and rotation of the
stars and may outshine the signal of dark matter annihilation. Only in a few
specific scenarios both requests are satisfied and the compact objects are dark
matter burners. The observational challenges are discussed and a possible way
to identify the dark matter burners is outlined.
