Uniqueness of the differential Mueller matrix of uniform homogeneous media

We show that the differential matrix of a uniform homogeneous medium containing birefringence may not be uniquely determined from its Mueller matrix, resulting in the potential existence of an infinite set of elementary polarization properties parameterized by an integer parameter. The uniqueness depends on the symmetry properties of a special differential matrix derived from the eigenvalue decomposition of the Mueller matrix. The conditions for the uniqueness of the differential matrix are identified, physically discussed, and illustrated in examples from the literature.