  The field equations of noncommutative gravity can be obtained by replacing
all exterior products by twist-deformed exterior products in the action
functional of general relativity, and are here studied by requiring that the
torsion 2-form should vanish, and that the Lorentz-Lie-algebra- valued part of
the full connection 1-form should be self-dual. Other two conditions,
expressing self-duality of a pair 2-forms occurring in the full curvature
2-form, are also imposed. This leads to a systematic solution strategy, here
displayed for the first time, where all parts of the connection 1-form are
first evaluated, hence the full curvature 2-form, and eventually all parts of
the tetrad 1-form, when expanded on the basis of {\gamma}-matrices. By assuming
asymptotic expansions which hold up to first order in the noncommutativity
matrix in the neighbourhood of the vanishing value for noncommutativity, we
find a family of self-dual solutions of the field equations. This is generated
by solving first a inhomogeneous wave equation on 1-forms in a classical curved
spacetime (which is itself self-dual and solves the vacuum Einstein equations),
subject to the Lorenz gauge condition. In particular, when the classical
undeformed geometry is Kasner spacetime, the above scheme is fully computable
out of solutions of the scalar wave equation in such a Kasner model.
