  In some recent papers some sufficiency conditions for the occurrence of a
$\mathbb{Z}_2$-symmetry breaking phase transition ($\mathbb{Z}_2$-SBPT) have
been showed starting from geometric-topological concepts of potential energy
landscapes. In particular, a $\mathbb{Z}_2$-SBPT can be triggered by
double-well potentials, or in an equivalent way, by dumbbell-shaped
equipotential surfaces. In this paper we introduce two models with a
$\mathbb{Z}_2$-SBPT which, due to their essential feature, show in the clearest
way the generating-mechanism of a $\mathbb{Z}_2$-SBPT above mentioned. These
models, despite they cannot be considered physical models, have all the
features of such models with the same kind of SBPT. At the end of the paper,
the $\phi^4$ model is revisited in the light of this approach. In particular,
the landscape of one of the model introduced here is turned out to be
equivalent to that of the mean-field $\phi^4$ model in a simplified version.
