  We consider finite-dimensional nonlinear systems with 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 for existence of the
families of nonlinear modes. We pay particular attention to the situations when
the underlying linear PT-symmetric operator is characterized by the presence of
degenerate eigenvalues or 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 for the system to admit one or
several integrals of motion, which turn out to be determined by the
pseudo-Hermiticity of the nonlinear operator.
