Completely integrable systems are Hamiltonian systems with “enough” first integrals and were originally introduced to model the phase spaces of mechanical systems with symmetries. Although the local structure of these systems is well understood, their global structure is far from being understood, particularly in dimensions greater than or equal to six.

In this talk, I will give a geometric introduction to completely integrable systems and present a six-dimensional system for which we have established the existence of an A_2 singularity, a type of singularity that had not been observed before in this setting. I will give a complete description of the topology of its singular fibers and explain how Hamiltonian monodromy can be studied for this system. I will conclude with some perspectives arising from this work, which is joint with K. Efstathiou, P. Mardešić, and D. Sugny.

This talk is based on the preprint “Hamiltonian monodromy in a Tavis–Cummings system with an A_2 singularity”, available on arXiv: https://arxiv.org/abs/2604.14835