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Generalized $μ$-$τ$ symmetry and discrete subgroups of O(3)

The generalized $μ$-$τ$ interchange symmetry in the leptonic mixing matrix $U$ corresponds to the relations: $|U_{μi}|=|U_{τi}|$ with $i=1,2,3$. It predicts maximal atmospheric mixing and maximal Dirac CP violation given $θ_{13} \neq 0$. We show that the generalized $μ$-$τ$ symmetry can arise if the charged lepton and neutrino mass matrices are invariant under specific residual symmetries contained in the finite discrete subgroups of $O(3)$. The groups $A_4$, $S_4$ and $A_5$ are the only such groups which can entirely fix $U$ at the leading order. The neutrinos can be (a) non-degenerate or (b) partially degenerate depending on the choice of their residual symmetries. One obtains either vanishing or very large $θ_{13}$ in case of (a) while only $A_5$ can provide $θ_{13}$ close to its experimental value in the case (b). We provide an explicit model based on $A_5$ and discuss a class of perturbations which can generate fully realistic neutrino masses and mixing maintaining the generalized $μ$-$τ$ symmetry in $U$. Our approach provides generalization of some of the ideas proposed earlier in order to obtain the predictions, $θ_{23}=π/4$ and $δ_{\rm CP} = \pm π/2$.