= Solution
Probability enters state $n$ through a birth from $n-1$ or a pair-death event from $n+2$, and leaves through either event at state $n$. Thus the <Kramers-Moyal master equation> is
$$
\boxed{
\frac{\partial P(n,t)}{\partial t}
=\lambda P(n-1,t)+\beta(n+2)^2P(n+2,t)
-(\lambda+\beta n^2)P(n,t)
},
$$
with probabilities and impossible transition rates taken as zero outside the nonnegative states.
A birth changes $n$ by $+1$, while a death event changes it by $-2$. Applying these increments to the first moment gives
$$
\boxed{
\frac d{dt}\langle n\rangle
=\lambda-2\beta\langle n^2\rangle
}.
$$
Consequently every steady state must satisfy
$$
\boxed{\langle n^2\rangle=\frac{\lambda}{2\beta}}.
$$
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