  In this paper, the second in a series, we continue to study the generalized
Lam\'{e} equation with the Treibich-Verdier potential \begin{equation*}
y^{\prime \prime }(z)=\bigg[ \sum_{k=0}^{3}n_{k}(n_{k}+1)\wp(z+\tfrac{
\omega_{k}}{2}|\tau)+B\bigg] y(z),\quad n_{k}\in \mathbb{Z}_{\geq0}
\end{equation*} from the monodromy aspect. We prove the existence of a
pre-modular form $Z_{r,s}^{\mathbf{n}}(\tau)$ of weight $\frac{1}{2}\sum
n_k(n_k+1)$ such that the monodromy data $(r,s)$ is characterized by
$Z_{r,s}^{\mathbf{n}}(\tau)=0$. This generalizes the result in \cite{LW2},
where the Lam\'{e} case (i.e. $n_1=n_2=n_3=0$) was studied by Wang and the
third author. As applications, we prove among other things that the following
two mean field equations \[\Delta u+e^u=16\pi\delta_{0}\quad\text{and}\quad
\Delta u+e^u=8\pi\sum_{k=1}^3\delta_{\frac{\omega_k}{2}}\] on a flat torus
$E_{\tau}:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau)$ has the same number of even
solutions. This result is quite surprising from the PDE point of view.
