  We analyse the MSSM parameter space and discuss the narrow band near the
so-called co-annihilation region where sleptons may be long-lived particles.
This region is consistent with the WMAP restrictions on the Dark matter and
depends on the value of $\tan\beta$. In this region staus are long-lived and
may go through the detector. Due to a relatively small mass (150 $\div$ 850
GeV) their production cross-section at LHC may reach a few % pb.
