  The paper contains constructions of Hilbert systems for the action of the
circle group $T$ using subgroups of implementable Bogoljubov unitaries w.r.t.
Fock representations of the Fermion algebra for suitable data of the selfdual
framework: ${\cal H}$ is the reference Hilbert space, $\Gamma$ the conjugation
and $P$ a basis projection on ${\cal H}.$ The group $C({spec} {\cal Z}\to T)$
of $T$-valued functions on ${spec} {\cal Z}$ turns out to be isomorphic to the
stabilizer of ${\cal A}$. In particular, examples are presented where the
center ${\cal Z}$ of the fixed point algebra ${\cal A}$ can be calculated
explicitly.
