  We discuss Lomonosov's proof of the Pontryagin-Krein Theorem on invariant
maximal non-positive subspaces, prove the refinement of one theorem from
\cite{OShT} on common fixed points for a group of fractional-linear maps of
operator ball and deduce its consequences. Some Burnside-type counterparts of
the Pontryagin-Krein Theorem are also considered.
