  In a recent experiment Lauber et al. have deformed cyclically a microwave
resonator and have measured the adiabatic normal-mode wavefunctions for each
shape along the path of deformation. The nontrivial observed cyclic phases
around a 3-fold degeneracy were accounted for by Manolopoulos and Child within
an approximate theory. However, open-path geometrical phases disagree with
experiment. By solving exactly the problem, we find unsuspected extra
degeneracies around the multiple one that account for the measured phase
changes throughout the path. It turns out that proliferation of additional
degeneracies around a multiple one is a common feature of quantum mechanics.
