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: ${\mathcal H}$ is the reference Hilbert space, $\Gamma$ the conjugation and $P$ a basis projection on ${\mathcal H}.$ The group $C(\text{spec} {\mathcal Z}\rightarrow T)$ of $T$-valued functions on $\text{spec} {\mathcal Z}$ turns out to be isomorphic to the stabilizer of ${\mathcal A}$. In particular, examples are presented where the center ${\mathcal Z}$ of the fixed point algebra ${\mathcal A}$ can be calculated explicitly.