On extended symmetries for the Galileon
  We investigate a large class of infinitesimal, but fully nonlinear in the
field, transformations of the Galileon and search for extended symmetries. The
transformations involve powers of the coordinates $x$ and the field $\pi$ up to
any finite order $N$. Up to quadratic order the structure of these symmetry
transformations is the unique generalisation of both the infinitesimal version
of the standard Galileon shift symmetry as well as a recently discovered
infinitesimal extension of this symmetry. The only higher-order extensions of
this symmetry we recover are (`Galileon dual' versions of) symmetries of the
standard kinetic term.
