  For a bounded planar domain $\Omega^0$ whose boundary contains a number of
flat pieces $\Gamma_i$ we consider a family of non-symmetric billiards $\Omega$
constructed by patching several copies of $\Omega^0$ along $\Gamma_i$'s. It is
demonstrated that the length spectrum of the periodic orbits in $\Omega$ is
degenerate with the multiplicities determined by a matrix group $G$. We study
the energy spectrum of the corresponding quantum billiard problem in $\Omega$
and show that it can be split in a number of uncorrelated subspectra
corresponding to a set of irreducible representations $\alpha$ of $G$. Assuming
that the classical dynamics in $\Omega^0$ are chaotic, we derive a
semiclassical trace formula for each spectral component and show that their
energy level statistics are the same as in standard Random Matrix ensembles.
Depending on whether ${\alpha}$ is real, pseudo-real or complex, the spectrum
has either Gaussian Orthogonal, Gaussian Symplectic or Gaussian Unitary types
of statistics, respectively.
