  By using the $L_e$ - $L_\mu$ - $L_\tau$ symmetry, we construct an
$SU(3)_L\times U(1)_N$ gauge model that provides two-loop radiative neutrino
masses as well as one-loop radiative neutrino masses. The generic smallness of
two-loop neutrino masses leading to $\Delta m^2_\odot$ compared with one-loop
neutrino masses leading to $\Delta m^2_{atm}$ successfully explains $\Delta
m^2_{atm}$ $>>$ $\Delta m^2_{\odot}$ by invoking the $L_e$ - $L_\mu$ - $L_\tau$
breaking. The Higgs scalar ($h^+$) that initiates radiative mechanisms is
unified into a Higgs triplet together with the standard Higgs scalar ($\phi^+$,
$\phi^0$) to form ($\phi^+$, $\phi^0$, $h^+$), which calls for three families
of lepton triplets: ($\nu^i_L$, $\ell^i_L$, $\omega^i_L$) (i = 1,2,3), where
$\omega^i$ denote heavy neutral leptons. The two-loop radiative mechanism is
found possible by introducing a singly charged scalar, which couples to
$\ell^i_R\omega^j_R$ (i,j = 2,3).
