= Solution
There is now no separate rate-$\nu$ clockwise transition. The only transitions are
$$
(m,n)\longrightarrow(m-1,n+1)\quad\text{at rate }\lambda_m,
\qquad
(m,n)\longrightarrow(m+1,n-1)\quad\text{at rate }\mu\quad(n>0).
$$
Both preserve $m+n$ modulo $M$. The initial state therefore selects the <closed communicating class> $m+n\equiv0\pmod M$, on which $m\equiv-n\pmod M$. The population is a <birth-death process> with birth rate $\lambda_{-n}$ and death rate $\mu$.
Define $a_0=1$ and $a_r=\mu^{-r}\prod_{j=0}^{r-1}\lambda_{-j}$ for $1\leq r<M$. The <detailed balance for a birth-death process> weights are
$$
w_n=\mu^{-n}\prod_{j=0}^{n-1}\lambda_{-j}.
$$
Writing $n=kM+r$ with $0\leq r<M$, the product condition gives $w_n=a_r\rho^{kM}$. Summing the resulting <geometric series> yields
$$
\boxed{\pi(m,kM+r)=\mathbf1_{\{m+r\equiv0\pmod M\}}\frac{(1-\rho^M)a_r\rho^{kM}}{\sum_{s=0}^{M-1}a_s}.}
$$
This is also the law in part (b) conditioned on $m+n\equiv0\pmod M$: the ratio of its weights at consecutive accessible populations is $\lambda_{-n}/\mu$. The stability condition is $\mu>\nu$, which uses the product of the periodic birth rates; it does not require $\mu>\lambda_m$ for every $m$.
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