[en] We have investigated the nuclear-structure details of the cross sections for the elastic scattering of Lightest Supersymmetric Particles (LSPs) from the promising dark-matter detectors ⁷¹Ga, ⁷³Ge and ¹²⁷I. The associated LSP detection sensitivities have been obtained by a folding procedure for several recently proposed SUSY models with different scalar and axial-vector characteristics. For the nuclear problem, a realistic microscopic Hamiltonian has been used within realistic model spaces. The diagonalization of this Hamiltonian has been done by using the Microscopic Quasiparticle-Phonon Model (MQPM), suitable for description of spectroscopic properties of medium-heavy and heavy odd-even nuclei. The formalism has been designed to incorporate in a trivial fashion parameter sets of any other SUSY-model scenarios