  Let $\mathbb B^n$ be the unit ball in $\mathbb C^n$ and $\mathbb H^n$ be the
homogeneous Siegel domain of the second kind which is biholomorphic to $\mathbb
B^n$. We show that the K\"ahler potential of $\mathbb H^n$ is unique up to the
automorphisms among K\"ahler potentials whose differentials have constant
norms. As an application, we consider a domain $\Omega$ in $\mathbb C^n$, which
is biholomorphic to $\mathbb B^n$. We show that if $\Omega$ is affine
homogeneous, then it is affine equivalent to $\mathbb H^n$. Assume next that
its canonical potential with respect to the K\"ahler--Einstein metric has a
differential with a constant norm. If the biholomorphism between $\Omega$ and
$\mathbb B^n$ is a restriction of a M\"obius transformation, then the map is
affine equivalent to a Cayley transform.
