Christoffel equation in the polarization variables

We formulate the classic Christoffel equation in the polarization variables and solve it for the slowness vectors of plane waves corresponding to a given unit polarization vector. Our analysis shows that, unless the equation degenerates and yields an infinite number of different slowness vectors, the finite nonzero number of its legitimate solutions varies from 1 to 4. Also we find a subset of triclinic solids in which the polarization field can have holes; there exist finite-size solid angles of polarization directions unattainable to any plane wave.