= Solution
For a <stellar polytrope> of index $n>0$, write $P=K\rho^{1+1/n}$ and combine <hydrostatic equilibrium> with <mass conservation> to obtain
$$
\frac1{r^2}\frac d{dr}\left(\frac{r^2}{\rho}\frac{dP}{dr}\right)=-4\pi G\rho.
$$
Introduce <Lane-Emden variables for a stellar polytrope>, $\rho=\rho_c\theta^n$, $P=P_c\theta^{n+1}$ and $r=\alpha\xi$, where
$$
\alpha^2=\frac{(n+1)P_c}{4\pi G\rho_c^2}.
$$
Since $\rho^{-1}dP/dr=(n+1)P_c\theta'/\rho_c\alpha$, the mechanical equation reduces to the <Lane-Emden equation>
$$
\boxed{\frac1{\xi^2}\frac d{d\xi}\left(\xi^2\frac{d\theta}{d\xi}\right)=-\theta^n,\qquad\theta(0)=1,\quad\theta'(0)=0.}
$$
A constant-density interior corresponds to the formal <polytrope of index zero>, with $\rho=\rho_c\theta^0=\rho_c$ where $\theta>0$. At $n=0$, regularity first gives $\xi^2\theta'=-\xi^3/3$, and a second integration gives
$$
\boxed{\theta(\xi)=1-\frac{\xi^2}{6},\qquad\xi_1=\sqrt6,\qquad R=\alpha\sqrt6.}
$$
Here $P=P_c\theta=P_c(1-r^2/R^2)$ and $\alpha^2=P_c/(4\pi G\rho_c^2)$, reproducing $P_c=2\pi G\rho_c^2R^2/3$. The <mass density> jumps from its constant interior value to zero at the surface, while <pressure> vanishes continuously. The relation $P=K\rho^{1+1/n}$ is singular at $n=0$; the regular dimensionless <pressure>/<mass density> formulation defines this incompressible structural limit. It does not mean that the gas's perturbative <stellar adiabatic exponent> is infinite: the hydrostatic ideal-gas toy model and its adiabatic response are distinct choices.
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