Homoclinic Bifurcations in Symmetric Unfoldings of a Singularity with Threefold Zero Eigenvalue

In this paper we study the singularity at the origin with threefold zero eigenvalue for symmetric vector fields with nilpotent linear part and 3jet Cequivalent to yx*zy*ax2yz with a 0. We first obtain several subfamilies of the symmetric versal unfoldings of this singularity by using the normal form and blowup methods under some conditions, and derive the local and global bifurcation behavior, then prove analytically the existence of the Sil'nikov homoclinic bifurcation for some subfamilies of the symmetric versal unfoldings of this singularity, by using the generalized Mel'nikov methods of a homoclinic orbit to a hyperbolic or nonhyperbolic equilibrium in a highdimensional space.