Summary: We consider a generalization of the 1:1:2 resonant swing-spring [see H. Dullin, A. Giacobbe and R. H. Cushman, Physica D 190, No. 1–2, 15–37 (2004; Zbl 1098.70520)] which is suggested both by the symmetries of this system and by its physical and in particular molecular realizations [see R. H. Cushman, H. R. Dullin, A. Giacobbe, D. D. Holm, M. Joyeux, P. Lynch, D. A. Sadovskiĭ and B.I. Zhilinskiĭ, Phys. Rev. Lett. 93, 024302-1–024302-4 (2004)]. Our generic integrable system is detuned off the exact Fermi resonance 1:2. The three-dimensional (3D) image of its energy–momentum map \(\mathcal{EM}\) consists either of two or three qualitatively different non-intersecting 3D regions: a regular region at low vibrational excitation, a region with monodromy similar to that studied for the exact resonance, and in some cases–an intermediate region in which the 3D set of regular values of \(\mathcal{EM}\) is partially self-overlapping while remaining connected. In the presence of this latter region, the system has an interesting property which we called bidromy. We analyze monodromy and bidromy for a concrete integrable classical Hamiltonian system of three coupled oscillators and for its quantum analog. We also show that the bifurcation involved in the transition from the regular region to the region with monodromy can be regarded as a special resonant equivariant analog of the Hamiltonian Hopf bifurcation.