= Solution
Put $f=1-2m/r$, use <proper time> $s$, and denote its <derivatives> by dots. The <timelike geodesic> equations, away from the Schwarzschild coordinate singularity, are
$$
\boxed{\begin{aligned}
\ddot t+\frac{f'}f\dot r\dot t&=0,\\
\ddot r+\frac{ff'}2\dot t^2-\frac{f'}{2f}\dot r^2-rf(\dot\theta^2+\sin^2\theta\dot\phi^2)&=0,\\
\ddot\theta+\frac{2\dot r}r\dot\theta-\sin\theta\cos\theta\dot\phi^2&=0,\\
\ddot\phi+\frac{2\dot r}r\dot\phi+2\cot\theta\dot\theta\dot\phi&=0,
\end{aligned}}
$$
with <normalization> $f\dot t^2-f^{-1}\dot r^2-r^2\dot\theta^2-r^2\sin^2\theta\dot\phi^2=1$. <Spherical symmetry> permits an <equatorial plane>, $\theta=\pi/2$, and the cyclic coordinates give $\boxed{E=f\dot t,\quad J=r^2\dot\phi}$. <Normalization> then gives $\dot r^2=E^2-f(1+J^2/r^2)$.
For $J\ne0$ and $u=m/r$, $du/d\phi=-m\dot r/J$. Thus
$$
\boxed{(u')^2=\frac{(E^2-1)m^2}{J^2}+\frac{2m^2}{J^2}u-u^2+2u^3.}
$$
At $E=1$, $J=4m$, the right side factors as $2u(u-1/4)^2$. Set $q=2\sqrt u$, so $u=q^2/4$ and $(q')^2=(q^2-1)^2/8$. Integrating the regions $0<q<1$ and $q>1$ gives, up to angular shifts and time reversal,
$$
\boxed{u=\tfrac14\tanh^2\!\left(\frac{\phi-\phi_0}{2\sqrt2}\right),\qquad
u=\tfrac14\coth^2\!\left(\frac{\phi-\phi_0}{2\sqrt2}\right).}
$$
The constant $u=1/4$, or $r=4m$, is the corresponding circular timelike orbit. The hyperbolic-tangent branch lies at $r>4m$ and approaches that orbit from outside as $|\phi|\to\infty$; the hyperbolic-cotangent branch lies at $0<r<4m$ and approaches it from inside. Near the limit $u-1/4$ decays exponentially as $e^{-|\phi|/\sqrt2}$. Since $\dot\phi$ tends to $J/(4m)^2=1/(4m)$, the limiting approach also takes infinite <proper time>. Neither nonconstant branch reaches and joins the <circular orbit> at a finite angular parameter.
Differentiating the orbit equation gives $u''=m^2/J^2-u+3u^2$. A perturbation about $u=1/4$ obeys $\delta u''=\delta u/2$, showing that the <circular orbit> is unstable. The two explicit curves are its <separatrix> branches, not ordinary small stable oscillations. For $E=1$ the energy is marginally bound, so the outer critical branch is the limiting orbit between <scattering> and <Schwarzschild marginally bound capture> from infinity. Time-reversed inner and outer branches describe the corresponding approach or departure from that unstable orbit. These are the <Explicit separatrices of the marginally bound Schwarzschild orbit>.
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