Abstract. A classical problem in the study of the (conservative) unfoldings of the so called Hopf-zero bifurcation, is the computation of the splitting of a heteroclinic connection which exists in the symmetric normal form along the z-axis. In this paper we derive the inner system associated to this singular problem, which is independent on the unfolding parameter. We prove the existence of two solutions of this system related with the stable and unstable manifolds of the unfolding, and we give an asymptotic formula for their difference. We check that the results in this work agree with the ones obtained in the regular case by the authors. Dedicated to Carles Simó for his 60th birthday 1. Introduction. W
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