= Solution
For $j\geq1$, the only incoming transition is from $j-1$, so a <stationary distribution> must satisfy
$$
\pi_j=\pi_{j-1}\frac\lambda j=\pi_0\frac{\lambda^j}{j!}.
$$
Normalization gives $\pi_0e^\lambda=1$, hence
$$
\boxed{\pi_j=e^{-\lambda}\frac{\lambda^j}{j!},\qquad j\geq0}.
$$
The zero-state balance also holds: $\sum_i\pi_i(1-\lambda/(i+1))=1-\sum_{j\geq1}\pi_j=\pi_0$. Thus the Poisson distribution is indeed stationary, including when $\lambda=1$. Irreducibility gives uniqueness.
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