  We study the effect of gradual symmetry breaking in a non-integrable system
on the level fluctuation statistics. We consider the case when the symmetry is
represented by a quantum number that takes one of two possible values, so that
the unperturbed system has a spectrum composed of two independent sequences.
When symmetry-breaking perturbation is represented by a random matrix with an
adjustable strength, the shape of the spectrum monotonously evolves towards the
Wigner distribution as the strength parameter increases. This contradicts the
observed behaviour of the acoustic resonance spectra in quartz blocks during
the breaking of a point-group symmetry that has two eigenvalues, where the
system changes in the beginning towards the Poisson statistics then turns back
to the GOE statistics. This behaviour is explained by assuming that the
symmetry breaking perturbation removes the degeneracy of a limited number of
levels, thus creating a third chaotic sequence. As symmetry breaking increases,
the new sequence grows at the expense of the initial pair until it overwhelms
the whole spectrum when the symmetry completely disappears. The calculated
spacing distribution and spectral rigidity are able to describe the evolution
of the observed acoustic resonance spectra.
