  BMPV black holes in flat transverse space and in Taub-NUT space have
identical near horizon geometries but different microscopic degeneracies. It
has been proposed that this difference can be accounted for by different
contribution to the degeneracies of these black holes from hair modes, --
degrees of freedom living outside the horizon. In this paper we explicitly
construct the hair modes of these two black holes as finite bosonic and
fermionic deformations of the black hole solution satisfying the full
non-linear equations of motion of supergravity and preserving the supersymmetry
of the original solutions. Special care is taken to ensure that these solutions
do not have any curvature singularity at the future horizon when viewed as the
full ten dimensional geometry. We show that after removing the contribution due
to the hair degrees of freedom from the microscopic partition function, the
partition functions of the two black holes agree.
