Local symmetry and the dependence on extended spacetime

We show that linearised E theory possesses a local symmetry at low levels provided the parameters of the local symmetry obey differential conditions that restrict their dependence on the extended spacetime. In the decomposition of E theory that leads to Siegel theory, also known as Double Field theory, we also find the analogous restrictions on the parameters. They are different to the section conditions which are universally used in this context. We also show that the dilaton equation of Siegel theory is invariant under the local symmetry if the parameters satisfy an analogous non-linear constraint on the parameters. We argue that there is no need to impose conditions on the fields of E theory or Siegel theory.