In the case $b_{1}(D)\neq 0$, from the proof we can make the same conclusion except the holomorphic line bundles $L_{+}$ and $L_{-}$ can not be prescribed as isomorphic to the powers on the given holomorphic line bundle $L$. Instead, as can be seen in the above proof, they are determined by the restriction of $\partial_{z}\tilde{\omega}(z)$ on the two ends. However, as pointed in Remark 4.3.2, we always have $L_{+}=L^{-\otimes k_{+}}\otimes\mathcal{F}$ and $L_{-}=L^{\otimes k_{-}}\otimes\mathcal{F}^{-1}$ for some holomorphic line bundle $\mathcal{F}$ on $D$ with $c_{1}(\mathcal{F})=0$. In particular the tensor product $L_{+}\otimes L_{-}$ is always isomorphic to $L^{k}$. The freedom of $\mathcal{F}$ corresponds exactly to the choice of the connection 1-form $\Theta$ in the construction of $\mathcal{M}^{*}$.