  Regular sequences of narrow peaks have been observed in the missing mass
spectra in the reactions pp --> p pi^+ X and pd --> ppX_1 below pion-production
threshold. They are interpreted in the literature as manifestations of
supernarrow light dibaryons, or nucleon resonances, or light pions forming
resonance states with the nucleon in its ground state. We discuss how existence
of such exotic states would affect properties of nuclear systems. We show that
the neutron star structure is drastically changed in all three cases. We find
that in the presence of dibaryons or nucleon resonances the maximal possible
mass of a neutron star would be smaller than the observational limit. Presence
of light pions does not contradict the observed neutron star masses. Light
pions allow for the existence of extended nuclear objects of arbitrary size,
bound by strong and electromagnetic forces.
