  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.
