  Obstruction flatness of a strongly pseudoconvex hypersurface $\Sigma$ in a
complex manifold refers to the property that any (local) K\"ahler-Einstein
metric on the pseudoconvex side of $\Sigma$, complete up to $\Sigma$, has a
potential $-\log u$ such that $u$ is $C^\infty$-smooth up to $\Sigma$. In
general, $u$ has only a finite degree of smoothness up to $\Sigma$. In this
paper, we study obstruction flatness of hypersurfaces $\Sigma$ that arise as
unit circle bundles $S(L)$ of negative Hermitian line bundles $(L, h)$ over
K\"ahler manifolds $(M, g).$ We prove that if $(M,g)$ has constant Ricci
eigenvalues, then $S(L)$ is obstruction flat. If, in addition, all these
eigenvalues are strictly less than one and $(M,g)$ is complete, then we show
that the corresponding disk bundle admits a complete K\"ahler-Einstein metric.
Finally, we give a necessary and sufficient condition for obstruction flatness
of $S(L)$ when $(M, g)$ is a K\"ahler surface $(\dim M=2$) with constant scalar
curvature.
