Isometries in the symmetrized bidisc, II

We prove that every map $f:U\to \text{GG}$ preserving the Poincar'e distance and the Kobayashi distance is holomorphic or anti-holomorphic, where $U$ is a connected open subset of the unit disc. We also study the nonroyal automorphism orbits of $\text{GG}$. A $C^1$ map on a connected relatively open part of such an orbit which preserves the restriction of the ambient Kobayashi–Royden metric is the restriction of a global automorphism or anti-automorphism of $\text{GG}$. The same conclusion holds for maps preserving the ambient Kobayashi distance.