= Solution
A statistic is sufficient when the conditional law of the sample given it is parameter-free; it is minimal sufficient when it is a function of every sufficient statistic. The likelihood factors as
$$
L=\bigl[\pi+(1-\pi)e^{-\lambda}\bigr]^{n_0}
(1-\pi)^{n-n_0}e^{-\lambda(n-n_0)}\lambda^S
\prod_i\frac1{X_i!},
$$
so the factorization theorem makes $(n_0,S)$ sufficient. The likelihood-ratio criterion shows that the ratio for two samples is parameter-independent exactly when both $n_0$ and $S$ agree. Thus
$$
\boxed{(n_0,S)\text{ is minimal sufficient}.}
$$
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