Resonant two-to-one oscillator coupling
= Resonant two-to-one oscillator coupling
For oscillators of frequencies $1$ and $2$, the perturbation
$$
H'=A^{\dagger2}B+A^2B^\dagger
$$
preserves $N_A+2N_B$. On the energy-$9/2$ basis $|3,0\rangle,|1,1\rangle$, it is the matrix
$$
\begin{pmatrix}0&\sqrt6\\\sqrt6&0\end{pmatrix},
$$
so the symmetric and antisymmetric combinations are exact eigenstates with shifts $\pm\sqrt6\lambda$.