How to embed an arbitrary Hamiltonian dynamics in a superintegrable (or just integrable) Hamiltonian dynamics

Given an arbitrary (autonomous) Hamiltonian , where the N components p n of the N -vector are the canonical momenta, the N components q n of the N -vector are the corresponding canonical coordinates and N is an arbitrary positive integer, we show how to manufacture (autonomous) Hamiltonians , featuring the N + 1 canonical momenta and the corresponding canonical coordinates , and having the following two properties: (i) the generic solutions of are periodic —entailing that the dynamics yielded by this Hamiltonian is (maximally) superintegrable , namely, it features 2 N + 1 functionally independent constants of motion, N + 1 of which in involution. (ii) On the manifold characterized by the condition , the coordinates and evolve trivially, and , while the evolution of the 2 N coordinates is that determined by the ( arbitrary !) Hamiltonian . This is related to an earlier finding by Bolsinov and Taimanov.