the symmetric sector, that is those states $\Psi$, which satisfy $\Psi(\lambda_1,\lambda_2,v,\phi) 
= \Psi(|\lambda_1|,|\lambda_2|,|v|,\phi)$. This allows to restrict the studies just to the positive octant $\lambda_1,\lambda_2,v>0$, on which an action of $\Theta_{\operatorname*{B1}}$ is given by (quite complicated) Eqs. $(3.35)$-$(3.37)$ of [@awe]. Its important feature is, that, analogously to the isotropic one, it divides $\Hil_{\phy}$ onto the superselection sectors built of the states supported just on the sets $\{(\lambda_1,\lambda_2,v); \lambda_1,\lambda_2\in\re, v\in\lat^+_{\varepsilon}\}$ with $\lat^+_{\varepsilon} := \{v=\varepsilon+4n;\ n\in\natu\}$, preserved by an action of $\Theta_{\operatorname*{B1}}$. Therefore to extract the physics one can consider just one of those sectors. Here, for simplicity we choose $\varepsilon=0$.

As we show in Appendix \[app:b1-sadj\] the operator $\Theta_{\operatorname*{B1}}$ admits self-adjoint extensions. Knowing its action one can in principle find the physical Hilbert space(s) corresponding to the model by analyzing the spectral properties of the extensions. On the other hand there exists a well defined procedure of the averaging over anisotropies (defined in [@awe] to build an embedding of the isotropic model in the homogeneous anisotropic one). In this article we will focus just on the space of averaged states $\bar{\Hil}_{\phy}$ and their physical properties.

### The isotropic sector

Following [@awe] and the ideas of [@bh-pert] we consider a projection ${\hat{\boldsymbol{P}}}$ 