  We characterize the existence of proper holomorphic mappings in the special
class of bounded $(1,2,...,n)$-balanced domains in $\mathbb{C}^n$, called the
symmetrized ellipsoids. Using this result we conclude that there are no
non-trivial proper holomorphic self-mappings in the class of symmetrized
ellipsoids. We also describe the automorphism groupof these domains.
