  We discuss an extended $SU(2)\times U(1)$ model which naturally leads to mass
scales and mixing angles relevant for understanding both the solar and
atmospheric neutrino anomalies. No right-handed neutrinos are introduced in the
model.The model uses a softly broken $L_e-L_{\mu}-L_{\tau}$ symmetry. Neutrino
masses arise only at the loop level. The one-loop neutrino masses which arise
as in the Zee model solve the atmospheric neutrino anomaly while breaking of
$L_e-L_{\mu}-L_{\tau}$ generates at two-loop order a mass splitting needed for
the vacuum solution of the solar neutrino problem. A somewhat different model
is possible which accommodates the large-angle MSW resolution of the solar
neutrino problem.
