  We consider the ``standard'' spectrum of the active neutrinos (characterized
by strong mass hierarchy and small mixing) with additional sterile neutrino,
$\nu_s$. The sterile neutrino mixes strongly with the muon neutrino, so that
$\nu_{\mu} \leftrightarrow \nu_s$ oscillations solve the atmospheric neutrino
problem. We show that the parametric enhancement of the $\nu_{\mu}
\leftrightarrow \nu_s$ oscillations occurs for the high energy atmospheric
neutrinos which cross the core of the Earth. This can be relevant for the
anomaly observed by the MACRO experiment. Solar neutrinos are converted both to
$\nu_{\mu}$ and $\nu_s$. The heaviest neutrino ($\approx \nu_{\tau}$) may
compose the hot dark matter of the Universe. Phenomenology of this scenario is
elaborated and crucial experimental signatures are identified. We also discuss
properties of the underlying neutrino mass matrix.
