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 = − ρ , where p r is the radial pressure, ρ being energy density. We find in a closed form the metric for an arbitrary equation of state for tangential pressure p θ ( ρ ). 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.