  A systematic study of deformations of four-dimensional Einsteinian
space-times embedded in a pseudo-Euclidean space $E^N$ of higher dimension is
presented. Infinitesimal deformations, seen as vector fields in $E^N$, can be
divided in two parts, tangent to the embedded hypersurface and orthogonal to
it; only the second ones are relevant, the tangent ones being equivalent to
coordinate transformations in the embedded manifold.
  The geometrical quantities can be then expressed in terms of embedding
functions $z^A$ and their infinitesimal deformations $v^A z^A \to {\tilde{z}}^A
= z^A + \epsilon v^A$. The deformations are called Einsteinian if they keep
Einstein equations satisfied up to a given order in $\epsilon$. The system so
obtained is then analyzed in particular in the case of the Schwarzschild metric
taken as the starting point, and some solutions of the first-order deformation
of Einstein's equations are found.
  We discuss also second and third order deformations leading to wave-like
solutions and to the departure from spherical symmetry towards an axial one
(the approximate Kerr solution)
