  X-ray observations have unveiled the existence of a family of radio-quiet
Isolated Neutron Stars whose X-ray emission is purely thermal, hence dubbed
X-ray Dim Isolated Neutron Stars (XDINSs). While optical observations have
allowed to relate the thermal emission to the neutron star cooling and to build
the neutron star surface thermal map, IR observations are critical to pinpoint
a spectral turnover produced by a so far unseen magnetospheric component, or by
the presence of a fallback disk. The detection of such a turnover can provide
further evidence of a link between this class of isolated neutron stars and the
magnetars, which show a distinctive spectral flattening in the IR.
  Here we present the deepest IR observations ever of five XDINSs, which we use
to constrain a spectral turnover in the IR and the presence of a fallback disk.
The data are obtained using the ISAAC instrument at the VLT.
  For none of our targets it was possible to identify the IR counterpart down
to limiting magnitudes H = 21.5 - 22.9. Although these limits are the deepest
ever obtained for neutron stars of this class, they are not deep enough to rule
out the existence and the nature of a possible spectral flattening in the IR.
We also derive, by using disk models, the upper limits on the mass inflow rate
in a fallback disk. We find the existence of a putative fallback disk
consistent (although not confirmed) with our observations.
