Do we have examples of a contractible **bounded** open set $D\subseteq\mathbf{C}^n$ such that $Hol(D)$ (the group of biholomorphisms $f:D\rightarrow D$) acts transitively on $D$ but such that there exists no symmetry at a given point $x\in D$ (so at all points by homogeneity). By a symmetry at $x$ I mean an element $s\in Hol(D)$ such that in a small neighborhood of $x$ only $x$ is fixed and $s^2=1$. 



A bounded homogeneous space which fails to be symmetric?
There are some examples given by Pjateckiĭ-Šapiro in
 Classification of bounded homogeneous regions in n-dimensional complex space. Dokl. Akad. Nauk SSSR 141 1961 316–319. 
and
On bounded homogeneous domains in an n-dimensional complex space. Izv. Akad. Nauk SSSR Ser. Mat. 26 1962 107–124. 


(I have a vague memory that the smallest examples are 4-dimensional, but might have misremembered.)


