  We show that the linear symplectic and anti-symplectic transformations form
the maximal covariance group for both the Wigner transform and Weyl operators.
The proof is based on a new result from symplectic geometry which characterizes
symplectic and anti-symplectic matrices, and which allows us, in addition, to
refine a classical result on the preservation of symplectic capacities of
ellipsoids.
