Splitting of Three Nearly Mass-Degenerate Neutrinos

Assuming the canonical seesaw mechanism together with an SO(3) family symmetry for leptons, broken only by the charged-lepton masses, I show that the three neutrinos of Majorana mass $m_0$ are split radiatively in two loops by a maximum finite calculable amount of order $10^{-9} m_0$. This is very suitable for dark matter and vacuum solar neutrino oscillations. I also discuss how atmospheric neutrino oscillations can be incorporated.