  Thin gravitating defects with conical singularities in higher codimensions
and with generalized Israel matching conditions are known to be inconsistent
for generic energy-momentum. A way to remove this inconsistency is proposed and
is realized for an axially symmetric gravitating codimension-two defect in six
dimensional Einstein gravity. By varying with respect to the brane embedding
fields, alternative matching conditions are derived, which are generalizations
of the Nambu-Goto equations of motion of the defect, consistent with bulk
gravity. For a maximally symmetric defect the standard picture is recovered.
The four-dimensional perfect fluid cosmology coincides with conventional FRW in
the case of radiation, but for dust it has rho^{4/3} instead of rho. A
four-dimensional black hole solution is presented having the Schwarzschild form
with a short-distance correction r^{-2}.
