Barotropic hydrostatic first integral (${\int }^p\frac{d\tilde{p}}{\rho (\tilde{p})+\tilde{p}}+\log N=C$)

For a barotropic fluid in hydrostatic equilibrium along a Killing vector, assume $\rho +p>0$. Integrating $dp/(\rho +p)=-d\log N$ gives the displayed first integral, constant on each connected fluid region. For the equation of state $p=w\rho$ with constant $w>0$, it gives $\rho N^{(1+w)/w}=C_1$.