  We propose a next-to-minimal supersymmetric Standard Model (NMSSM) extended
by an $\mathbb{A}_{4}\times \boldsymbol{Z}_{3}$ flavor symmetry and three
right-handed neutrinos providing a detailed study of the neutrino sector and a
solution to the domain wall problem. In this proposal, neutrino masses are
generated through Type I seesaw mechanism while the mixing angles are described
by the trimaximal mixing realized using the NMSSM singlet S and only two flavon
fields. The phenomenology of neutrino parameters is studied for normal and
inverted mass hierarchies. In particular, we numerically evaluated the
observables related to neutrino masses and mixing, namely, $\sum m_{i}$,
$m_{ee}$, $m_{\nu_{e}}$, and $\delta_{CP}$ where we find that the ranges of
$m_{ee}$ and $m_{\nu_{e}}$ are accessible by current and future experiments
while the obtained ranges of $\sum m_{i}$ and $\delta_{CP}$ lie within the
current experimental data. Another attractive feature we discussed in this
paper is the circumvention of the domain wall problem induced by the
spontaneous breaking of the $\mathbb{A}_{4}\times \boldsymbol{Z}_{3}$ discrete
symmetry. We first showed that the domain walls in the charged lepton sector
occur at high energy scale leading to unproblematic domain walls, while in the
neutrino sector they are inevitable. Then, to solve this problem, we
reconsidered the well-known approach that relies on the explicit breaking of
the discrete symmetry through the insertion of Planck-suppressed operators
induced by supergravity.
  \keywords{Neutrino physics, Discrete flavor symmetry,Trimaximal mixing,
Domain walls}
