Spherically symmetric accretion
Let me again point you the the textbook by Vladimir Karas;
Section 4.1. The reader is alsoencouraged to go through the introduction of Section 4.
My handnotes are also available
online, although I think they do not add anything to the textbook.
Let me emphasise a few points:

equation (4.6) is also known as Parker's equation and is used to describe a basic model of
stellar winds (see also lecture
Astrophysics II)

logical arguments are used to determine qualitative structure of a subset
of solutions of Parker's equation; exact solutions are dependent on boundary conditions
and the particular form of the equation of state and can be obtained only by means of
numerical integration (except for some singular cases)

the exact coordinates of the sonic point,
${r}_{\mathsf{s}}$,
can be obtained only through integration
of the Parker's equation. Instead, characteristic accretion radius,
${r}_{\mathsf{acc}}\approx GM\u2215{c}_{\mathsf{s}}^{2}\left(\infty \right)$,
can be obtained directly from parameters of the system and it is commonly used as an estimate
of the radius of region in which gravity of the central
object dominates motion of the fluid and from which the fluid is typically accreted
onto it in real systems (which typically do not obey the asymptotic boundary conditions
considered in simple, spherically symmetric model)