  We study several classes of non-Hermitian Hamiltonian systems, which can be
expressed in terms of bilinear combinations of Euclidean Lie algebraic
generators. The classes are distinguished by different versions of antilinear
(PT)-symmetries exhibiting various types of qualitative behaviour. On the basis
of explicitly computed non-perturbative Dyson maps we construct metric
operators, isospectral Hermitian counterparts for which we solve the
corresponding time-independent Schroedinger equation for specific choices of
the coupling constants. In these cases general analytical expressions for the
solutions are obtained in the form of Mathieu functions, which we analyze
numerically to obtain the corresponding energy eigenspectra. We identify
regions in the parameter space for which the corresponding spectra are entirely
real and also domains where the PT symmetry is spontaneously broken and
sometimes also regained at exceptional points. In some cases it is shown
explicitly how the threshold region from real to complex spectra is
characterized by the breakdown of the Dyson maps or the metric operator. We
establish the explicit relationship to models currently under investigation in
the context of beam dynamics in optical lattices.
