  This is part of a series of papers on the inverse spectral problem for
bounded analytic plane domains. Here, we use the trace formula established in
the first paper (`Balian-Bloch trace formula') to explicitly calculate wave
trace invariants associated to bouncing ball orbits and dihedral rays. We use
these invariants to prove that simply connected bounded analytic plane domains
with one symmetry (which reverses a bouncing ball orbit of fixed length L) are
spectrally determined in this class.
