Stationary modes and integrals of motion in nonlinear lattices with a PT-symmetric linear part

We consider finite-dimensional nonlinear systems with a linear part described by a parity-time (PT-) symmetric operator. We investigate bifurcations of stationary nonlinear modes from the eigenstates of the linear operator and consider a class of PT-symmetric nonlinearities allowing the existence of families of nonlinear modes. We pay particular attention to situations when the underlying linear PT-symmetric operator is characterized by the presence of degenerate eigenvalues or an exceptional-point singularity. In each of the cases we construct formal expansions for small-amplitude nonlinear modes. We also report a class of nonlinearities allowing the system to admit one or several integrals of motion, which turn out to be determined by the pseudo-hermiticity of the nonlinear operator.