A model for real-space singlet pairs, hopping on the quasi 2D antiferr
omagnetic background of localized spins is further analysed. It is sho
wn that lattice vibrations (if any), which break the symmetry of the i
n-plane oxygen position with respect to the nearest neighboring copper
sites, may strongly influence both the magnitude and the temperature
dependence of the O-17 NMR relaxation rate. For the Y site a situation
is analogous, although the different modes may contribute to the effe
ct. A relation of the results obtained to the Millis-Monien-Pines anal
ysis is discussed. The validity of the mean-field random phase approxi
mation for the description of the 2D antiferromagnet used previously i
s analysed. It is shown that the 2D fluctuations of the phase of the (
bond) mean field parameter lead to the power-law decay with the distan
ce of the corresponding correlation function (which remains finite in
the mean field approximation). The spectrum of the phase fluctuations
is found to be propagative in the one-loop approximation. An interacti
on of the real-space singlet pairs with the long-wavelength phase fluc
tuations is discussed.