  The possibility of the extension of spatial diffeomorphisms to a larger
family of symmetries in a class of classical field theories is studied. The
generator of the additional local symmetry contains a quadratic kinetic term
and a potential term which can be a general (not necessarily local) functional
of the metric. From the perspective of the foundation of Einstein's gravity our
results are positive: The extended constraint algebra is either that of
Einstein's gravity, or ultralocal gravity. If our goal is a simple modification
of Einstein's gravity that for example makes it perturbatively renormalizable,
as has recently been suggested, then our results show that there is no such
theory within this class.
