Integrable and Superintegrable Klein-Gordon and Schr\"odinger Type Dimers

A $PT$-symmetric dimer is a two-site nonlinear oscillator or a nonlinear Schr\"odinger dimer where one site loses and the other site gains energy at the same rate. We present a wide class of integrable oscillator type dimers whose Hamiltonian is of arbitrary even order. Further, we also present a wide class of integrable and superintegrable nonlinear Schr\"odinger type dimers where again the Hamiltonian is of arbitrary even order.