  We classify, up to local unitary equivalence, local unitary stabilizer Lie
algebras for symmetric mixed states into six classes. These include the
stabilizer types of the Werner states, the GHZ state and its generalizations,
and Dicke states. For all but the zero algebra, we classify entanglement types
(local unitary equivalence classes) of symmetric mixed states that have those
stabilizers. We make use of the identification of symmetric density matrices
with polynomials in three variables with real coefficients and apply the
representation theory of SO(3) on this space of polynomials.
