  Continuous $U(1)_{L_\mu-L_\tau}$ symmetry can generate quasi degenerate mass
spectrum for both left handed light and right handed heavy Majorana neutrinos
assuming that the symmetry preserving non zero parameters are nearly same.
There is an accidental $\mu\tau$ exchange symmetry in the light and heavy
neutrino Majorana mass terms. This implies $\theta_{13}=0$ and
$\theta_{23}=\frac{\pi}{4}$. In addition it generates another zero mixing angle
and one zero mass difference. We restrict ourselves to type-I See-Saw mechanism
for generation of light neutrino mass. We have found that under
$U(1)_{L_\mu-L_\tau}$ symmetry cosmological lepton asymmetry vanishes. We break
$U(1)_{L_\mu-L_\tau}$ such a way that the $\mu\tau$ exchange symmetry preserves
in the neutrino sector. We have seen that light neutrino phenomenology can be
explained under soft breaking of this symmetry. We have observed that softness
of this symmetry breaking depends on the degeneracy of the light neutrino mass
spectrum. Quasi-degeneracy of right handed neutrino mass spectrum opens an
option for resonant leptogenesis. The degeneracy of the right handed neutrino
mass spectrum is restricted through light neutrino data. We observed that for
generation of right sized baryon asymmetry common neutrino mass scale $m_0$
have to be of the order of $\sqrt{\Delta m^2_{\rm atm}}$ and corresponding
right handed neutrino mass scale have to be nearly $10^{13}$ GeV. We also have
discussed the effect of RG evolution on light neutrino spectrum and also on
baryon asymmetry.
