Inverted equation of state and general approach to vacuum-like configurations

We consider spherically symmetric static black hole configurations that obey the vacuum equation of state: $p_r=-\rho$, where $p_r$ is the radial pressure, $\rho$ being energy density. We find in a closed form the metric for an arbitrary equation of state for tangential pressure $p_{\theta }(\rho )$. The corresponding formulas enable us to embrace compact Schwarzschild-like configurations and dispersed systems. They include metrics with a regular center and singular ones. In a particular case, the metric of the Kiselev black hole is reproduced.