Deviation from $\mu -\tau$ Symmetry with New Approaches

We propose two new predictive neutrino mass matrix textures that deviate from $\mu$-$\tau$ symmetry and explore their phenomenological implications. These textures are significant due to their predictive nature and the unique approach they introduce for breaking the $\mu$-$\tau$ symmetry. Both textures predict six physical parameters and impose sharp constraints on ${\theta }_{23}$ and $\delta$, considering both normal and inverted neutrino mass orderings. In addition, we realize the textures in their exact form within the framework of the seesaw mechanism, based on the $SU(2{)}_L\times A_4\times Z_3$ symmetry group.