The symmetry groups of bifurcations of integrable Hamiltonian systems

Two-dimensional atoms are investigated; these are used to code bifurcations of the Liouville foliations of nondegenerate integrable Hamiltonian systems. To be precise, the symmetry groups of atoms with complexity at most 3 are under study. Atoms with symmetry group $\mathbb Z_p\oplus\mathbb Z_q$ are considered. It is proved that $\mathbb Z_p\oplus\mathbb Z_q$ is the symmetry group of a toric atom. The symmetry groups of all nonorientable atoms with complexity at most 3 are calculated. The concept of a geodesic atom is introduced.
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