  A distinguished (invariant) Bondi-like coordinate system is defined in the
spacetime neighbourhood of a non-expanding horizon of arbitrary dimension via
geometry invariants of the horizon. With its use, the radial expansion of a
spacetime metric about the horizon is provided and the free data needed to
specify it up to given order are determined in spacetime dimension $4$. For the
case of an electro-vacuum horizon in $4$-dimensional spacetime the necessary
and sufficient conditions for the existence of a Killing field at its
neighbourhood are identified as differential conditions on the horizon data and
data on null surface transversal to the horizon.
