  Parameter dependent non-Hermitian quantum systems typically not only possess
eigenvalue degeneracies, but also degeneracies of the corresponding
eigenfunctions at exceptional points. While the effect of two coalescing
eigenfunctions on cyclic parameter variation is well investigated, little
attention has hitherto been paid to the effect of more than two coalescing
eigenfunctions. Here a characterisation of behaviours of symmetric Hamiltonians
with three coalescing eigenfunctions is presented, using perturbation theory
for non-Hermitian operators. Two main types of parameter perturbations need to
be distinguished, which lead to characteristic eigenvalue and eigenvector
patterns under cyclic variation. A physical system is introduced for which both
behaviours might be experimentally accessible.
