The solution of Einstein field equations in case of static spherically symmetric perfect fluid sphere is given by the metric $$ds^2=f^2(r,\alpha)~c^2 dt^2-g^2(r,\alpha) dr^2-r^2 (d\phi^2+\sin^2{\theta}~d \theta^2) \tag{1}.$$ Null-geodesics in such spacetime are defined by the condition $ds=0$. In terms of affine parameter the geodesic curve $r(\lambda)$ is given implicitly as $$\lambda=A \int{f(r,\alpha)\cdot}g(r,\alpha)~dr+B,\tag{2}$$ where the dimensionless curvature radius $r$ and the parameter $\alpha$ are defined as $r~\hat{=}~r/R$ and $\alpha~=~r_s/R$, with $R$ and $r_s$ curvature radius of fluid and Schwarzschild sphere correspondingly. In case of constant energy density fluid sphere (Schwarzschild interior solution) the equation (2) yields \begin{equation} \label{affine lambda} \lambda/A =\left\{ \begin{array}{rcl} 3/2 \sqrt{1/\alpha-1}~ \arcsin{(\sqrt{\alpha}~r)}-r/2~ & \mbox{for} & 0\leq r \leq 1 \\ \\r-1 +3/2 \sqrt{1/\alpha-1}~ \arcsin{(\sqrt{\alpha}~1)}-1/2~ & \mbox{for} & 1 > r \leq \infty~. \end{array}\right.\tag{3} \end{equation} The constant $A$ remains undetermined as a scaling factor.
For $\alpha<8/9$ (Buchdahl limit) the geodesics of the spacetime (1) should be complete, i.e. their affine parameter $\lambda$ should extend from $-\infty$ to $+\infty$. However, due to equation (3) the range of affine parameter is from $0$ to $+\infty$. Thus, every null geodesics is a half line and all of them are in- or outgoing geodesics. To resolve that problem (I need a straight line crossing the center) I have thought that in order to get the full strait line I could use the idea of antipodal identification for the pairs of starting points ($r=0$) and combining it with a sign change of the affine parameter.
Would it be mathematically correct?
Remark
Be aware that $r$ is not a radial coordinate of spherical system but the Gaussian curvature radius of a 2-sphere. It cannot be negative.