Four-neutrino mass spectra and the Super-Kamiokande atmospheric up-down asymmetry

In the framework of schemes with mixing of four massive neutrinos, which can accommodate the atmospheric, solar and LSND ranges of $\Delta m^2$ and contain three active neutrinos and a sterile one, we show that, in the whole region of $\Delta m_{\mathrm{LSND}}^2$ allowed by LSND, the Super-Kamiokande up-down asymmetry excludes all mass spectra with a group of three close neutrino masses separated from the fourth mass by the LSND gap of order 1 eV. Only two schemes with mass spectra in which two pairs of close masses are separated by the LSND gap can describe the Super-Kamiokande up-down asymmetry and all other existing neutrino oscillation data.