= Solution
Use the $C$ equation at level $i-1$ to express $h_{C_i}$ in terms of the preceding level, then substitute it into the $W$ equation at level $i$. This yields
$$
\boxed{h^{(i)}=Ah^{(i-1)},\qquad A=\begin{pmatrix}
1+\alpha/\lambda&-\alpha/\lambda\\
\dfrac{\beta(1+\alpha/\lambda)}{\mu+\beta}&\dfrac{\mu-\beta\alpha/\lambda}{\mu+\beta}
\end{pmatrix}.}
$$
The standard eigenpair is $\boxed{1,\ (1,1)^T}$. Each row sums to one because constant functions are harmonic for every Markov generator. This recurrence matrix need not have nonnegative entries; it encodes a harmonic equation, rather than a transition matrix. The superscript $(i)$ labels the level vector and is not an additional printed subpart.
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