  We obtain the general $n(\ge 4)$-dimensional static solution with an
$(n-2)$-dimensional Einstein base manifold for a perfect fluid obeying a linear
equation of state $p=-(n-3)\rho/(n+1)$. It is a generalization of Semiz's
four-dimensional general solution with spherical symmetry and consists of two
different classes. Through the Buchdahl transformation, the class-I and
class-II solutions are dual to the topological Schwarzschild-Tangherlini-(A)dS
solution and one of the $\Lambda$-vacuum direct-product solutions,
respectively. While the metric of the spherically symmetric class-I solution is
$C^\infty$ at the Killing horizon for $n=4$ and $5$, it is $C^1$ for $n\ge 6$
and then the Killing horizon turns to be a parallelly propagated curvature
singularity. For $n=4$ and $5$, the spherically symmetric class-I solution can
be attached to the Schwarzschild-Tangherlini vacuum black hole with the same
value of the mass parameter at the Killing horizon in a regular manner, namely
without a lightlike massive thin-shell. This construction allows new
configurations of an asymptotically (locally) flat black hole to emerge. If a
static perfect fluid hovers outside a vacuum black hole, its energy density is
negative. In contrast, if the dynamical region inside the event horizon of a
vacuum black hole is replaced by the class-I solution, the corresponding matter
field is an anisotropic fluid and may satisfy the null and strong energy
conditions. While the latter configuration always involves a spacelike
singularity inside the horizon for $n=4$, it becomes a non-singular black hole
of the big-bounce type for $n=5$ if the ADM mass is larger than a critical
value.
