charge $+7$, see table \[particles\], allows to write down the term \[theta13op\] \^d\_2 N N/M . The VEV of $\Phi^d_2$, see Eq.(\[VEVdown\]), breaks the $Z_2$ symmetry generated by U = (

[cc]{} 0 & 1\
1 & 0

) in the chosen basis, because it is not an eigenvector of $U$. On the other hand, it trivially leaves invariant the $Z_2$ symmetry generated by $S$, since $S=\mathds{1}$ for the representation ${\bf 2}$. In this way a non-zero value of $\theta_{13}^l$ is generated in the neutrino sector, while tri-maximal mixing related to the generator $S$ is still maintained, see Eqs.(\[mnuLO\], \[Strimax\]).[^5] Knowing that $U$ is represented for ${\bf 3}$ and ${\bf 3'}$ by U = (

[ccc]{} 1 & 0 & 0\
0 & 0 & 1\
0 & 1 & 0

) , one immediately sees that the broken $Z_2$ symmetry is identified with $\mu
-\tau$ symmetry which protects $\theta_{13}^\nu$ as well as $\theta_{23}^\nu$ from deviating from their TB mixing values. As a consequence, we expect that the atmospheric mixing angle is corrected in a similar way to $\theta_{13}^\nu$. For $\theta_{13}^\nu \sim \la$ this implies a correction term proportional to $\la$ also for the atmospheric mixing angle. This expectation is 