  Diffraction of elastic waves is considered for a system consisting of two
parallel arrays of thin (subwavelength) cylinders that are arranged
periodically. The embedding media supports waves with all polarizations, one
longitudinal and two transverse, having different dispersion relations. An
interaction with scatters mixes longitudinal and one of the transverse modes.
It is shown that the system supports bound states in the continuum (BSC) that
have no specific polarization, that is, there are standing waves localized in
the scattering structure whose wave numbers lies in the first open diffraction
channels for both longitudinal and transverse modes. BSCs are shown to exists
only for specific distances between the arrays and for specific values of the
wave vector component along the array. An analytic solution is obtained for
such BSCs. For distances between the parallel arrays much larger than the
wavelength, BSCs is proved to exist due to destructive interference of the far
field resonance radiation, similar to the interference in a Fabry-Perot
interferometer, that can occur simultaneously for both propagating modes.
