  We derive the effective cosmological equations for a non-$\mathbb{Z}_2$
symmetric codimension one brane embedded in an arbitrary D-dimensional bulk
spacetime, generalizing the $D=5,6$ cases much studied previously. As a
particular case, this may be considered as a regularized codimension (D-4)
brane avoiding the problem of curvature divergence on the brane. We apply our
results to the case of spherical symmetry around the brane and to partly
compactified AdS-Schwarzschild bulks.
