Property (2).

For the second important property, if we assume that there is a radial position $\xi=\xi_{e}$, where the subscript $e$ stands for extremum, at which we have that $\pi'(\xi_{e})=0$ and $\pi(\xi_{e})\neq 0$, then there is a definite and unique solution to these conditions that presents itself, namely the expression within braces in Equation (37) must vanish, leading to

  $$4(n+1)\left[\xi-\gamma(\xi)\right]F(\xi,\pi) = n \left[1+F(\xi,\pi)\right] \left[ \gamma(\xi) + \xi \pi(\xi) F(\xi,\pi) \right].$$ (52)

Since we already know that $\gamma (\xi )$ must be a monotonically increasing function that is limited from above by $\xi_{M}$, the condition $\pi'(\xi)=0$ with $\pi(\xi)\neq 0$ identifies the inflection point $\xi_{e}$ of $\gamma (\xi )$, which is also the point of maximum of $\pi (\xi )$. The equation above then identifies in a definite way the value of $\gamma(\xi_{e})$ as a certain function of $\pi_{e}=\pi(\xi_{e})$ at that inflection point, a relation that is given explicitly by

  $$\gamma(\xi_{e}) = \xi_{e} F(\xi_{e},\pi_{e})\, \frac { 4(n... ..._{e}) \left[1+F(\xi_{e},\pi_{e})\right] } { n+(5n+4)F(\xi_{e},\pi_{e}) }.$$ (53)

This relation is the second important property of the solutions, which identifies the radial position $\xi_{e}$ of the inflection point. This constitutes a radial position and a pair of values $\gamma(\xi_{e})$ and $\pi(\xi_{e})$ of $\gamma (\xi )$ and $\pi (\xi )$ that can be used as the starting point for a numerical integration of the equation, to either one of the two sides of $\xi_{e}$. When there is a well-defined matter region, it allows us to perform the numerical integration starting from a regular point in the interior of the matter interval, rather than at one of the two ends of that interval which, as we will see shortly, are soft singular points.