  A theorem due to Cartan asserts that every origin-preserving automorphism of
bounded circular domains with respect to the origin is linear. In the present
paper, by employing the theory of Bergman's representative domain, we prove
that under certain circumstances Cartan's assertion remains true for
quasi-circular domains in $\mathbb C^n$. Our main result is applied to obtain
some simple criterions for the case $n=3$ and to prove that
Braun-Kaup-Upmeier's theorem remains true for our class of quasi-circular
domains.
