  The $\mathcal{O}(\partial^2)$ background independent flow equations for
conformally reduced gravity are shown to be equivalent to flow equations
naturally adapted to scalar field theory with a wrong sign kinetic term. This
sign change is shown to have a profound effect on the renormalization group
properties, broadly resulting in a continuum of fixed points supporting both a
discrete and a continuous eigenoperator spectrum, the latter always including
relevant directions. The properties at the Gaussian fixed point are understood
in particular depth, but also detailed studies of the local potential
approximation, and the full $\mathcal{O}(\partial^2)$ approximation are given.
These results are related to evidence for asymptotic safety found by other
authors.
