Quantization and fixed points of non-integrable Weyl theory

We consider a simple but generic model of gravity where Weyl--invariance is realized thanks to the presence of a gauge field for dilatations. We quantize the theory by suitably defining renormalization group flows that describe the integration of successive momentum shells, in such a way that Weyl--invariance is maintained in the flow. When the gauge fields are massless the theory has, in addition to Weyl invariance, an abelian gauge symmetry. According to the definition of the cutoff, the flow can break or preserve this extended symmetry. We discuss the fixed points of these flows.