Nice!

I've been a bit absorbed in blogging about this stuff and am just catching up with your work.

We can why there's a 1-parameter family of solutions of the master equation in this case using Noether's theorem. We've got a reversible reaction

$$A + 2 B \leftrightarrow 3 A$$

and if we write

$$A + 2 B = 3 A$$

we can solve and get

$$B = A$$

so the overall effect of this reaction is to turn an $A$ into a $B$. This means that the number of $A$'s, plus the number of $B$'s, is conserved. So, we get a conserved quantity

$$O = N_A + N_B = a_A^\dagger a_A + a_B^\dagger a_B$$

and we therefore can apply any function of $O$ to an equilibrium solution $\Psi$ of the master equation and get another equilibrium solution:

$$H \Psi = 0 \implies H (f(O) \Psi) = 0$$

since Noether's theorem says

$$[H , O] = 0$$

so

$$[H, f(O)] = 0$$

so

$$H f(O) \Psi = f(O) H \Psi = 0$$

This kind of argument is fairly general. We just need to find a conserved quantity!