  Recent results in the construction of anomaly-free models of loop quantum
gravity have shown obstacles when local physical degrees of freedom are
present. Here, a set of no-go properties is derived in polarized Gowdy models,
raising the question whether these systems can be covariant beyond a background
treatment. As a side product, it is shown that normal deformations in classical
polarized Gowdy models can be Abelianized.
