  We have analyzed a $U(1)$ model which is broken explicitly to a $Z_2$ model.
The model results in generating two stable domain walls, in contrast with the
more common $N_{DW}=1$ version which is prevalently used to explain axion
invisibility for $U_{PQ}(1)$ model. We have been meticulous to take into
account any possible relation with previous studies, even if they apparently
belong to different lines of research. Then we have scrutinized the domain wall
properties of the model,proposing a rigorous approximate solution which fully
satisfies boundary conditions and the static virial theorem simultaneously.
Invoking the mentioned approximation, we have been able to obtain an analytical
insight about the effect of parameters on the domain wall features,
particularly on their surface energy density which is of great importance in
cosmological studies when one tries to avoid domain wall energy domination
problem. Next, we have mainly focused on the likely inflationary scenarios of
the model, including saddle point inflation, again insisting on analytical
discussions to be able to follow the role of parameters. We have tried to
categorize any inflationary scenario into the known categories to take
advantage of the previous detailed studies under the inflationary topic over
the decades. We have concluded that any successful inflationary scenario
requires large fields definition for the model. Calculations are mainly done
analytically and numerical results are used as supportive material.
