= Solution
Regularity at the center requires $m(0)=0$ and a finite central density $\rho_c$. The <equation of state> fixes $p_c=p(\rho_c)$, and the regular central series begins
$$
m(r)=\frac{4\pi}{3}\rho_c r^3+O(r^5),
\qquad
p(r)=p_c+O(r^2).
$$
For each admissible $\rho_c$, ordinary-differential-equation uniqueness determines $m,p,\rho$ outward until the first zero of $p$, which defines $R$. The remaining additive constant in $\Phi$ is fixed by matching to the exterior Schwarzschild time coordinate. Thus smooth stars form a one-parameter family labelled uniquely by $\rho_c$.
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