Dynamically implementing the $\overline{\mu}$-scheme in cosmological and spherically symmetric models in an extended phase space model

We consider an extended phase space formulation for cosmological and spherically symmetric models in which the choice of a given $\overline{\mu}$-scheme can be implemented dynamically. These models are constructed in the context of the relational formalism by using a canonical transformation on the extended phase space which provides a Kucha\v{r} decomposition of the extended phase space. The resulting model can be understood as a gauge-unfixed model of a given $\overline{\mu}$-scheme. We use this formalism to investigate the restrictions to the allowed $\overline{\mu}$-scheme from this perspective and discuss the differences in the cosmological and spherically symmetric case. This method can be useful, for example, to obtain a $\overline{\mu}$-scheme in a top-down derivation from full LQG to symmetry reduced effective models, where for some models only the $\mu_0$-scheme has been obtained so far.