= Solution
With constant <isothermal sound speed> $a_T$, the equation of state is $p=a_T^2\rho$. Spherical <hydrostatic equilibrium> and <mass> continuity require
$$
a_T^2\frac{d\rho}{dr}=-\rho\frac{GM(r)}{r^2},\qquad\frac{dM}{dr}=4\pi r^2\rho.
$$
Try $\rho=A/r^2$. With no central point <mass>, $M(r)=4\pi Ar$, and hydrostatic balance becomes $2a_T^2A=4\pi GA^2$. The positive-<density> branch is therefore
$$
\boxed{A=\frac{a_T^2}{2\pi G},\qquad\rho=\frac{a_T^2}{2\pi Gr^2},\qquad M(r)=\frac{2a_T^2r}{G}.}
$$
The constant external <pressure> fixes the truncation <radius> $r_s$ through $p(r_s)=p_0$, giving
$$
\boxed{r_s=\frac{a_T^2}{\sqrt{2\pi Gp_0}},\qquad M_s=\frac{2a_T^2r_s}{G}.}
$$
This is the <pressure-truncated singular isothermal gas sphere>, one hydrostatic solution, not a claim that every isothermal cloud has a singular centre. Its central <density> diverges, but its enclosed <mass> tends to zero as $r\to0$.
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