  We propose a non-minimal left-right symmetric model (LRSM) with Parity
Symmetry where the fermion mixings arise as result of imposing an ${\bf
S}_{3}\otimes {\bf Z}_{2}$ flavor symmetry, and an extra ${\bf Z}^{e}_{2}$
symmetry is considered to suppress some Yukawa couplings in the lepton sector.
As a consequence, the effective neutrino mass matrix possesses approximately
the $\mu-\tau$ symmetry. The breaking of the $\mu-\tau$ symmetry induces
sizable non zero $\theta_{13}$, and the deviation of $\theta_{23}$ from
$45^{\circ}$ is strongly controlled by an $\epsilon$ free parameter and the
complex neutrino masses. Then, an analytic study on the extreme Majorana phases
is done since these turn out to be relevant to enhance or suppress the reactor
and atmospheric angle. So that we have constrained the parameter space for the
$\epsilon$ parameter and the lightest neutrino mass that accommodate the mixing
angles. The highlighted results are: a) the normal hierarchy is ruled out since
the reactor angle comes out being tiny, for any values of the Majorana phases;
b) for the inverted hierarchy there is one combination in the extreme phases
where the values of the reactor and atmospheric angles are compatible up to $2,
3~\sigma$ of C. L., but the parameter space is tight; c) the model favors the
degenerate ordering for one combination in the extreme Majorana phases. In this
case, the reactor and atmospheric angle are compatible with the experimental
data for a large set of values of the free parameters. Therefore, this model
may be testable by the future result that the Nova and KamLAND-Zen
collaborations will provide.
