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Spectral degeneracies of quantum magnets are often described as diabolicalpoints or magnetic Weyl points, which carry topological charge. Here, we studya simple, yet experimentally relevant quantum magnet: two localized interactingelectrons subject to spin-orbit coupling. In this setting, the degeneracies arenot necessarily isolated points, but can also form a line or a surface. Weidentify ten different possible geometrical patterns formed by these degeneracypoints, and study their stability under small perturbations of the Hamiltonian.Stable structures are found to depend on the relative sign of the determinantsof the two $g$-tensors, $\cal S$. Both for ${\cal S}=+1$ and ${\cal S}=-1$, twostable configurations are found, and three out of these four configurations areformed by pairs of Weyl points. These stable regions are separated by a surfaceof almost stable configurations, with a structure akin to co-dimension onebifurcations.

Magnetic degeneracy points in interacting two-spin systems: geometrical patterns, topological charge distributions, and their stability