= Solution
The angular Euler-Lagrange equations are homogeneous in $\dot\theta$ and $\dot\phi$. Therefore initial data with both angular velocities zero give the unique solution with constant $\theta$ and $\phi$, so a radial geodesic remains radial.
Time-translation symmetry supplies the <geodesic conserved quantity from a Killing vector>
$$
e=w\dot t.
$$
Metric compatibility makes the squared tangent norm another constant,
$$
k=g(\dot\gamma,\dot\gamma)
=-w\dot t^2+\frac{\dot r^2}{w}
=\boxed{-\frac{e^2}{1+r^2}+\frac{\dot r^2}{1+r^2}}.
$$
For a proper-time parametrized timelike geodesic $k=-1$, for a null geodesic $k=0$, and for a unit-speed spacelike geodesic $k=1$. The constant $e$ is the conserved energy per unit mass associated with the static Killing vector $\partial_t$.
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