Given an arbitrary (autonomous) Hamiltonian , where the N components pn of the N-vector are the canonical momenta, the N components qn 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 2N + 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 2N coordinates is that determined by the (arbitrary!) Hamiltonian . This is related to an earlier finding by Bolsinov and Taimanov.