  Positive results of dark matter searches in experiments DAMA/NaI and
DAMA/LIBRA confronted with results of other groups can imply nontrivial
particle physics solutions for cosmological dark matter. Stable particles with
charge -2, bound with primordial helium in O-helium "atoms" (OHe), represent a
specific nuclear-interacting form of dark matter. Slowed down in the
terrestrial matter, OHe is elusive for direct methods of underground Dark
matter detection using its nuclear recoil. However, low energy binding of OHe
with sodium nuclei can lead to annual variations of energy release from OHe
radiative capture in the interval of energy 2-4 keV in DAMA/NaI and DAMA/LIBRA
experiments. At nuclear parameters, reproducing DAMA results, the energy
release predicted for detectors with chemical content other than NaI differ in
the most cases from the one in DAMA detector. Moreover there is no bound
systems of OHe with light and heavy nuclei, so that there is no radiative
capture of OHe in detectors with xenon or helium content. Due to dipole Coulomb
barrier, transitions to more energetic levels of Na+OHe system with much higher
energy release are suppressed in the correspondence with the results of DAMA
experiments. The proposed explanation inevitably leads to prediction of
abundance of anomalous Na, corresponding to the signal, observed by DAMA.
