= Solution
For $X=(x,q)^T$,
$$
A=\begin{pmatrix}\mu\lambda&-\mu\\0&\nu\end{pmatrix},
\qquad
bb^T=\begin{pmatrix}0&0\\0&\alpha\nu\end{pmatrix}.
$$
The Lyapunov equation gives $\sigma_{qq}=\alpha/2$, $\sigma_{xq}=\lambda\sigma_{xx}$, and $(\mu\lambda+\nu)\sigma_{xq}=\mu\sigma_{qq}$. Therefore
$$
\boxed{\sigma=
\begin{pmatrix}
\dfrac{\mu\alpha}{2\lambda(\mu\lambda+\nu)}&
\dfrac{\mu\alpha}{2(\mu\lambda+\nu)}\\[8pt]
\dfrac{\mu\alpha}{2(\mu\lambda+\nu)}&\dfrac\alpha2
\end{pmatrix}}.
$$
Solved by gpt-5.6-sol high.
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