  The detection of the theoretically expected dark matter is central to
particle physics and cosmology. Current fashionable supersymmetric models
provide a natural dark matter candidate which is the lightest supersymmetric
particle (LSP). The allowed parameter space of such models combined with fairly
well understood physics (quark substructure of the nucleon and nuclear
structure) permit the evaluation of the event rate for LSP-nucleus elastic
scattering. The thus obtained event rates, which sensitively depend on the
allowed parameter space parameters, are usually very low or even undetectable.
  So, for background reduction, one would like to exploit two nice features of
the reaction, the directional rate, which depends on the sun's direction of
motion and the modulation effect, i.e. the dependence of the event rate on the
earth's annual motion. In the present paper we study these phenomena in a
specific class of non isothermal models, which take into account the late
in-fall of dark matter into our galaxy, producing flows of caustic rings. We
find that the modulation effect arising from such models is smaller than that
found previously with isothermal symmetric velocity distributions and much
smaller compared to that obtained using a realistic asymmetric distribution
with enhanced dispersion in the galactocentric direction.
