Exceptional points in quantum and classical dynamics

We notice that, when a quantum system involves exceptional points, i.e. the special values of parameters where the Hamiltonian loses its self-adjointness and acquires the Jordan block structure, the corresponding classical system also exhibits a singular behaviour associated with restructuring of classical trajectories. The system with the crypto-Hermitian Hamiltonian H = (p²+z²)/2 -igz⁵ and hyper-ellictic classical dynamics is studied in details. Analogies with supersymmetric Yang-Mills dynamics are elucidated