  The on lattice $\phi^4$ model is a paradigmatic example of continuous real
variables model undergoing a continuous symmetry braking phase transition
(SBPT). In this paper we study the $\mathbb{Z}_2$-symmetric mean-field version
without the quadratic term of the local potential. Obviously, the
simplification is directly extensible to the other symmetry groups for which
the model undergoes a SBPT. We show that the $\mathbb{Z}_2$-SBPT is not
affected by the quadratic term, and that the potential energy landscape turns
out greatly simplified. In particular, there exist only three critical points,
to confront with an amount growing as $e^N$ ($N$ is the number of degrees of
freedom) of the model with non-vanishing quadratic term. In our opinion, this
is an crucial feature because in recent years the study of the link between
statistical mechanic and geometric-topological properties of configuration
space has received an increasing attention. In this paper we study the
equipotential hypersurfaces with the aim of deepening our understanding of the
link between SBPTs and the truly essential geometric-topological properties of
the energy potential landscape.
