The metric of static spherically symmetric spacetime can be written as $$ds^2=e^{2\nu}~c^2 dt^2-e^{2\lambda}~dr^2-r^2~(d\phi^2+\sin^2{\theta}~d \theta^2) \tag{1},$$ where $r$ is curvature radius of a 2-sphere ($r\ge0$).
Defining dimensionless variable $r$ and parameter $\alpha$ $$ r\hat{=}\frac{r}{R},~~~r_S\equiv\frac{2 G M}{c^2},~~~\alpha\equiv\frac{r_S}{R},$$ the metric components of spacetime in question are
\begin{equation}
{\rm e}^{-\lambda} =\left\{ \begin{array}{rcl} \sqrt{1-\alpha~r^2}~ & \mbox{,} & 0\leq r \leq 1
\\
\\\sqrt{1-\alpha~ r^{-1}}~ & \mbox{,} & 1 < r < \infty~, \end{array}\right.\tag{2}
\end{equation}
and
\begin{equation}
{\rm e}^{\nu} =\left\{ \begin{array}{rcl} 3/2~\sqrt{1-\alpha}-1/2~\sqrt{1-\alpha~r^2})~ & \mbox{,} & 0\leq r \leq 1
\\
\\\sqrt{1-\alpha~ r^{-1}}~ & \mbox{,} & 1 < r < \infty~.\tag{3} \end{array}\right.
\end{equation}
The corresponding affine parameter for radial null geodesics in this spacetime, expressed implicitly, reads
\begin{equation}
\label{affine lambda}
\lambda(r,\theta,\phi) =\pm~ A~\left\{ \begin{array}{rcl} 3/2~\sqrt{1/\alpha-1}~ \arcsin{(\sqrt{\alpha}~r)}-1/2~r & \mbox{,} & 0 \leq r \leq 1
\\
\\r-1+3/2~\sqrt{1/\alpha-1}~ \arcsin{(\sqrt{\alpha}~1)}-1/2 & \mbox{,} & 1 < r < \infty~. \tag{4}\end{array}\right.
\end{equation}
The constant $A$ in the equation is a scale factor, the sign plus denotes outgoing, the minus ingoing geodesics.
Below the Buchdahl limit ($\alpha<8/9$) the geodesics should be complete, i.e. their affine parameter $\lambda$ should extend from $-\infty$ to $+\infty$ (see Valter Moretti, https://physics.stackexchange.com/a/637972).
However, due to equation (4) $\lambda$ extends only from $0$ to $+\infty$. Hence, every radial null geodesics is a half line. What am I missing here?
To resolve that problem I have thought of idea of antipodal identification that would equate the points of two half lines in combination with the sign switch of the affine parameter $$r(\lambda,\theta,\phi)\leftrightarrow r(-\lambda,\theta-\pi,\phi+\pi). \tag{5} $$ and thus representing full geodesics with affine parameter $\lambda \in (-\infty,+\infty)$
Would it be a right procedure?
