= Solution
Write the radial velocity as $u(r)$, allowing $u<0$ for <accretion flow>, and let $c^2=\gamma p/\rho$ be the squared <adiabatic sound speed>. The steady <continuity equation>, radial <Euler equations for an inviscid fluid> and <entropy advection equation> give
$$
\frac{d}{dr}(r^2\rho u)=0,\qquad
u\frac{du}{dr}=-\frac1\rho\frac{dp}{dr}-\frac{d\Phi}{dr},\qquad
u\frac{d}{dr}\left(\frac{p}{\rho^\gamma}\right)=0.
$$
On a nonzero smooth flow branch, the <polytropic equation of state> is $p=K\rho^\gamma$ with constant $K$. Thus the two useful first integrals for <spherically symmetric adiabatic flow> are
$$
\boxed{4\pi r^2\rho u=\mathcal J,\qquad
\frac{u^2}{2}+\frac{c^2}{\gamma-1}+\Phi=\mathcal B.}
$$
Here $\mathcal J$ is the signed mass flux and $\mathcal B$ is the <Bernoulli function>. The <specific enthalpy> is $c^2/(\gamma-1)$; the prescribed <Newtonian gravitational potential> has no contribution from the gas's own gravity. Eliminating the density derivative using <mass conservation> gives the differential form
$$
\boxed{\left(u-\frac{c^2}{u}\right)u'
=\frac{2c^2}{r}-\Phi'.}
$$
This <sonic point> equation displays both the singular coefficient and the numerator that must vanish for a smooth <transonic branch>.
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