Minisuperspace description of $f(Q)$-cosmology

We investigate the existence of minisuperspace description for the homogeneous cosmological field equations within the framework of symmetric teleparallel $f(Q)$-gravity. We consider the background space to be described by the isotropic Friedmann--Lema\^{\i}tre--Robertson--Walker geometry, the anisotropic Kantowski-Sachs and the anisotropic Bianchi III geometries. Across all these models, we establish that the field equations in $f(Q)$-cosmology exhibit second-order characteristics in the coincident gauge and those of a sixth-order theory in the non-coincident gauge. Specifically, within the latter scenario, the dynamic degrees of freedom are attributed to two scalar fields. Finally, as an example of integrability, we derive a vacuum cosmological solution within the non-coincident gauge.