The normalised H'enon-Heiles family exhibits a degenerate bifurcation when passing through the separable case "fi = 0". We clarify the relation between this degeneracy and the integrability at fi = 0 . Furthermore we show that the degenerate bifurcation carries over to the H'enon-Heiles family itself. x1 Introduction Originally motivated by galactic dynamics, the so-called H'enon-Heiles family of Hamiltonian systems in two degrees of freedom has gained a paradigmatic status. It served as a model in further applications to e.g. molecular dynamics or the (restricted) three body problem and was also studied in its own right. This family is defined by H(x 1 ; x 2 ; p 1 ; p 2 ) = 1 2 (p 2 1 + p 2 2 ) + ! 2 2 (x 2 1 + x 2 2 ) + " (ff x 3 1 3 + fi x 1 x 2 2 ) : (1) The pioneering work [H'enon,Heiles;64] contained numerical experiments on the case ff = 1 ; fi = \Gamma1, showing how "the integrability of the system breaks down" as the perturbation is switched on. Later studies, e.g. [..