  It is predicted that for sufficiently strong electron-phonon coupling an
anomalous quantum chaotic behavior develops in certain types of suspended
electro-mechanical nanostructures, here comprised by a thin cylindrical quantum
dot (billiard) on a suspended rectangular dielectric plate. The deformation
potential and piezoelectric interactions are considered. As a result of the
electron-phonon coupling between the two systems the spectral statistics of the
electro-mechanic eigenenergies exhibit an anomalous behavior. If the center of
the quantum dot is located at one of the symmetry axes of the rectangular
plate, the energy level distributions correspond to the Gaussian Orthogonal
Ensemble (GOE), otherwise they belong to the Gaussian Unitary Ensemble (GUE),
even though the system is time-reversal invariant.
