  We study theoretical models of three coupled wave guides with a
$\mathcal{PT}$-symmetric distribution of gain and loss. A realistic matrix
model is developed in terms of a three-mode expansion. By comparing with a
previously postulated matrix model it is shown how parameter ranges with good
prospects of finding a third-order exceptional point (EP3) in an experimentally
feasible arrangement of semiconductors can be determined. In addition it is
demonstrated that continuous distributions of exceptional points, which render
the discovery of the EP3 difficult, are not only a feature of extended wave
guides but appear also in an idealised model of infinitely thin guides shaped
by delta functions.
