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We analyze the generating mechanisms for heteroclinic cycles in$\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_2$--equivariant ODEs, notinvolving Hopf bifurcations. Such cycles have been observed in particle physicssystems 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 availabletheoretical data explaining these phenomena. We use singularity theory to studythe equivalence in the group-symmetric context, as well as the recognitionproblem for the simplest bifurcation problems with this symmetry group.Singularity results highlight different mechanisms for the appearance ofheteroclinic cycles, based on the transition between the bifurcating branches.On the other hand, we analyze the heteroclinic cycle of a generic dynamicalsystem with the symmetry of the group$\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_2$ acting on aeight--dimensional torus $\mathbb{T}^8,$ constructed via a Cayley graph, underweak coupling. We identify the conditions for heteroclinic cycle between fourequilibria in the three--dimensional fixed point subspaces of some of theisotropy subgroups of$\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{S}^1.$ We alsoanalyze the stability of the heteroclinic cycle.

Heteroclinic Cycles in Systems with $\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_2$ Symmetry, Revisited