  The Seiberg--Witten map is a powerful tool in non-commutative field theory,
and it has been recently obtained in the literature for gravity itself, to
first order in non-commutativity. This paper, relying upon the pure-gravity
form of the action functional considered in Ref. 2, studies the expansion to
first order of the non-commutative Einstein equations, and whether the
Seiberg--Witten map can lead to a solution of such equations when the
underlying classical geometry is Schwarzschild.
