The general static spherical perfect fluid solution with EoS parameter w=-1/5

The general analytical solution for the static spherically symmetric metric supported by a perfect fluid with isothermal (proportional) equation-of-state $p = w \rho$ is not known at the time of this writing, except for the trivial cases $w=0$ and $w=-1$; and for $w=-1/3$. We show that if Buchdahl coordinates are used, the problem becomes analytically solvable for $w=-1/5$; display and discuss the solution(s), and exhibit the connection of this case to the $w=-1$ case.