According to the classical approach to physical interpretation of multiple roots of the Christoffel equation in the theory of elastic wave propagation in anisotropic media, a sum of isonormal plane waves propagating in the direction of acoustic axes can have either arbitrary or circular polarization. The main question posed in this paper is as follows: can one apply conclusions drawn for plane waves to other phenomena, in particular, to waves generated by a point source? The paper proposes a new principle of physical interpretation of degeneracies stating that any assessment of the polarizations (and the group velocities) of waves propagating in anisotropic media is reasonable if there exists an experiment with a point source in which the assessment agrees with the general symmetry of the experiment. From the viewpoint of this interpretation, all degeneracies are considered on the wavefront. The inferences drawn from the performed analysis might appear surprising: in all considered cases of degeneracies (such as conic axes, tangent degeneracy on the symmetry axis of infinite order in transversely isotropic media, intersections of the slowness surfaces), ambiguity in determination of the polarization vectors either does not exist for any experiment or can be removed based on the symmetry of an experiment. A condition for convexity of the slowness surface of the fastest wave is formulated in this context
