Calogero, F.Gómez-Ullate Otaiza, DavidSantin, Paolo M.Sommacal, M2023-06-202023-06-202005-10-140305-447010.1088/0305-4470/38/41/004https://hdl.handle.net/20.500.14352/51453© IOP Publishing. It is a pleasure to acknowledge illuminating discussions with Boris Dubrovin, Yuri Fedorov, Jean-Pierre Fran¸coise, Fran¸cois Leyvraz, Jaume Llibre, Alexander Mikhailov and Carles Simó.We introduce and discuss a simple Hamiltonian dynamical system, interpretable as a three-body problem in the (complex) plane and providing the prototype of a mechanism explaining the transition from regular to irregular motions as travel on Riemann surfaces. The interest of this phenomenology-illustrating the onset in a deterministic context of irregular motions-is underlined by its generality, suggesting its eventual relevance to understand natural phenomena and experimental investigations. Here only some of our main findings are reported, without detailing their proofs: a more complete presentation will be published elsewhere.engjournal articlehttp://dx.doi.org/10.1088/0305-4470/38/41/004http://iopscience.iop.orghttp://arxiv.org/abs/nlin/0507024open access51-73Many-body problemSolvable dynamical-systemsPeriodic-solutionsDifferential-equationsHamiltonian-systemsOscillatorsPainleveNonintegrabilityIntegrabilityGaloreFísica-Modelos matemáticosFísica matemática