  We study the anisotropic Choquard--Pekar equation which de-scribes a polaron
in an anisotropic medium. We prove the uniqueness and non-degeneracy of
minimizers in a weakly anisotropic medium. In addition, for a wide range of
anisotropic media, we derive the symmetry properties of minimizers and prove
that the kernel of the associated linearized operator is reduced, apart from
three functions coming from the translation invariance, to the kernel on the
subspace of functions that are even in each of the three principal directions
of the medium.
