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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 $p=-(n-3)ρ/(n+1)$. 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 $Λ$-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.