  In this paper the Seiberg-Witten map is first analyzed for non-commutative
Yang-Mills theories with the related methods, developed in the literature, for
its explicit construction, that hold for any gauge group. These are exploited
to write down the second-order Seiberg-Witten map for pure gravity with a
constant non-commutativity tensor. In the analysis of pure gravity when the
classical space-time solves the vacuum Einstein equations, we find for three
distinct vacuum solutions that the corresponding non-commutative field
equations do not have solution to first order in non-commutativity, when the
Seiberg-Witten map is eventually inserted. In the attempt of understanding
whether or not this is a peculiar property of gravity, in the second part of
the paper, the Seiberg-Witten map is considered in the simpler case of Maxwell
theory in vacuum in the absence of charges and currents. Once more, no obvious
solution of the non-commutative field equations is found, unless the
electromagnetic potential depends in a very special way on the wave vector.
