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We consider integrable Hamiltonian systems in three degrees of freedom nearan elliptic equilibrium in 1:1:-2 resonance. The integrability originates fromaveraging along the periodic motion of the quadratic part and an imposedrotational symmetry about the vertical axis. Introducing a detuning parameterwe find a rich bifurcation diagram, containing three parabolas of HamiltonianHopf bifurcations that join at the origin. We describe the monodromy of theresulting ramified 3-torus bundle as variation of the detuning parameter letsthe system pass through 1:1:-2 resonance.

Bifurcations and Monodromy of the Axially Symmetric 1:1:-2 Resonance