  Many physical systems can be described by nonlinear eigenvalues and
bifurcation problems with a linear part that is non-selfadjoint e.g. due to the
presence of loss and gain. The balance of these effects is reflected in an
antilinear symmetry, like e.g. the PT-symmetry, of the problem. Under this
condition we show that the nonlinear eigenvalues bifurcating from real linear
eigenvalues remain real and the corresponding nonlinear eigenfunctions remain
symmetric. The abstract results are applied in a number of physical models of
Bose-Einstein condensation, nonlinear optics and superconductivity, and further
numerical analysis is performed.
