Generic three-parameter unfolding of a completely degenerate germ of a vector field on ${\mathbb{R}}^3$

In this work, we analyze a completely degenerate germ of a real vector field at the origin of ${\mathbb{R}}^3$. We construct a generic three-parameter unfolding of this germ and identify the associated local bifurcations. We prove that the phase portrait of the unfolded system contains a one-parameter family of heteroclinic orbits connecting two hyperbolic equilibrium points.