  For every mapping of a perturbed spacetime onto a background and with any
vector field $\xi$ we construct a conserved covariant vector density $I(\xi)$,
which is the divergence of a covariant antisymmetric tensor density, a
"superpotential". $ I(\xi)$ is linear in the energy-momentum tensor
perturbations of matter, which may be large; $I(\xi)$ does not contain the
second order derivatives of the perturbed metric. The superpotential is
identically zero when perturbations are absent.
  By integrating conserved vectors over a part $\Si$ of a hypersurface $S$ of
the background, which spans a two-surface $\di\Si$, we obtain integral
relations between, on the one hand, initial data of the perturbed metric
components and the energy-momentum perturbations on $\Si$ and, on the other
hand, the boundary values on $\di\Si$. We show that there are as many such
integral relations as there are different mappings, $\xi$'s, $\Si$'s and
$\di\Si$'s. For given boundary values on $\di\Si$, the integral relations may
be interpreted as integral constraints (e.g., those of Traschen) on local
initial data including the energy-momentum perturbations. Conservation laws
expressed in terms of Killing fields $\Bar\xi$ of the background become
"physical" conservation laws.
  In cosmology, to each mapping of the time axis of a Robertson-Walker space on
a de Sitter space with the same spatial topology there correspond ten
conservation laws. The conformal mapping leads to a straightforward
generalization of conservation laws in flat spacetimes. Other mappings are also
considered. ...
