Integrability and trajectory confinement in $\mathcal{P}\mathcal{T}$-symmetric waveguide arrays

We consider $\mathcal{P}\mathcal{T}$-symmetric ring-like arrays of optical waveguides with purely nonlinear gain and loss. Regardless of the value of the gain-loss coefficient, these systems are protected from spontaneous $\mathcal{P}\mathcal{T}$-symmetry breaking. If the nonhermitian part of the array matrix has cross-compensating structure, the total power in such a system remains bounded – or even constant – at all times. We identify two-, three-, and four-waveguide arrays with cross-compensatory nonlinear gain and loss that constitute completely integrable Hamiltonian systems.