The effect of some possible non standard WIMP velocity distributions, like the Debris Flows recently proposed, on the direct dark matter detection rates is investigated. We find that such distributions may be deciphered from the data, especially if t
he time variation of the event rates due to the annual motion of the Earth is observed
In the present paper we obtain the WIMP velocity distribution in our vicinity starting from spherically symmetric WIMP density profiles in a self consistent way by employing the Eddington approach. By adding a reasonable angular momentum dependent te
rm in the expression of the energy, we obtain axially symmetric WIMP velocity distributions as well. We find that some density profiles lead to approximate Maxwell-Boltzmann distributions, which are automatically defined in a finite domain, i.e. the escape velocity need not be put by hand. The role of such distributions in obtaining the direct WIMP detection rates, including the modulation, is studied in some detail and, in particular, the role of the asymmetry is explored.
