  The double copy is a map from non-abelian gauge theories to gravity, that has
been demonstrated both for scattering amplitudes and exact classical solutions.
In this study, we reconsider the double copy for exact solutions that are
self-dual in either the gauge or gravity theory. In this case, one may
formulate a general double copy in terms of a certain differential operator,
which generates the gauge and gravity solutions from a harmonic function
residing in a biadjoint scalar theory. As an illustration, we examine the
single copy of the well-known Eguchi-Hanson instanton in gravity. The gauge
field thus obtained represents an abelian-like object whose field is
dipole-like at large distances, and which has no magnetic or electric charge.
