  We consider a class of generally non-self-adjoint eigenvalue problems which
are nonlinear in the solution as well as in the eigenvalue parameter ("doubly"
nonlinear). We prove a bifurcation result from simple isolated eigenvalues of
the linear problem using a Lyapunov-Schmidt reduction and provide an expansion
of both the nonlinear eigenvalue and the solution. We further prove that if the
linear eigenvalue is real and the nonlinear problem $\mathcal P\mathcal
T$-symmetric, then the bifurcating nonlinear eigenvalue remains real. These
general results are then applied in the context of surface plasmon polaritons
(SPPs), i.e. localized solutions for the nonlinear Maxwell's equations in the
presence of one or more interfaces between dielectric and metal layers. We
obtain the existence of transverse electric SPPs in certain $\mathcal P\mathcal
T$-symmetric configurations.
