  We analyze the generating mechanisms for heteroclinic cycles in
$\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_2$--equivariant ODEs, not
involving Hopf bifurcations. Such cycles have been observed in particle physics
systems with the mentioned symmetry, in absence of the Hopf bifurcation, see
\cite{bury} and \cite{Park}, and as far as we know, there is no available
theoretical data explaining these phenomena. We use singularity theory to study
the equivalence in the group-symmetric context, as well as the recognition
problem for the simplest bifurcation problems with this symmetry group.
Singularity results highlight different mechanisms for the appearance of
heteroclinic cycles, based on the transition between the bifurcating branches.
On the other hand, we analyze the heteroclinic cycle of a generic dynamical
system with the symmetry of the group
$\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_2$ acting on a
eight--dimensional torus $\mathbb{T}^8,$ constructed via a Cayley graph, under
weak coupling. We identify the conditions for heteroclinic cycle between four
equilibria in the three--dimensional fixed point subspaces of some of the
isotropy subgroups of
$\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{S}^1.$ We also
analyze the stability of the heteroclinic cycle.
