Note that $\mathbb{C}^*$ can be interpreted as the space of flat $\mathbb{C}^*$-connections on the dual of $\mathbb{C}^*$. Our goal is to find a similar construction for $\mathbb{C}$, particularly in the following case.
Consider the universal centralizer of $\operatorname{PGL}_2(\mathbb{C})$, which can be defined as follows:
Let $\Sigma=\{\begin{bmatrix} 0 & u \\ 1 & 0 \end{bmatrix}\in \mathfrak{sl}_2\mathbb{C}\mid u\in \mathbb{C}\}$. Each $u\in\mathbb{C}$ corresponds to a regular adjoint orbit in $\mathfrak{sl}_2\mathbb{C}$.
For convenience, we will not distinguish between $u$ and $\begin{bmatrix} 0 & u \\ 1 & 0 \end{bmatrix}$.
The universal centralizer of $\operatorname{PGL}_2(\mathbb{C})$ is the following commuting variety given by: $$X=\{(u,g)\in \Sigma\times \operatorname{PGL}_2(\mathbb{C})\mid \operatorname{Ad}_g u=u\}$$
Let $\pi:X\rightarrow \Sigma $ be the projection to the first factor. Then for each $u\in \Sigma, \pi^{-1}(u)=:G_u=\{\begin{bmatrix} a & bu \\ b & a \end{bmatrix}\in \operatorname{PGL}_2(\mathbb{C})\mid a,b\in \mathbb{C}\}$ is the centralizer of $u$ in $\operatorname{PGL}_2(\mathbb{C})$. For $u\neq 0$, $\pi^{-1}(u)$ is a maximal torus in $\operatorname{PGL}_2(\mathbb{C})$, while $\pi^{-1}(0)\cong \mathbb{C}$ represents the centralizer of the regular nilpotent element $\begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix}$.
Notice that the Langlands dual group of $\operatorname{PGL}_2(\mathbb{C})$ is $\operatorname{SL}_2(\mathbb{C})$. For $u\neq 0$, denote the corresponding centralizer of $u$ in $\operatorname{SL}_2(\mathbb{C})$ by $\tilde{G}_u$. For $u\neq 0$, $\tilde{G}_u$ is the dual torus of $G_u$, meaning that $G_u=\operatorname{Hom}(\pi_1(\tilde{G}_u),\mathbb{C}^*)$, so $G_u$ can be viewed as the moduli space of flat $\mathbb{C}^*$-connections on $\tilde{G}_u$.
We are now interested in the moduli space of lagrangians with flat connections in $T^*\operatorname{SL}_2(\mathbb{C})\cong \mathfrak{sl}_2\times\operatorname{SL}_2(\mathbb{C})$, where the identification is given by killing form. Given $u\in \Sigma$, denote the corresponding adjoint orbit in $\mathfrak{sl}_2\mathbb{C}$ by $O_u$. Then define $\tilde{L}_u=\{(x,g)\in \mathfrak{sl}_2\times\operatorname{SL}_2(\mathbb{C})\mid x\in O_u, \operatorname{Ad}_gx=x\}$. We have the equivariant isomorphism $\tilde{L}_u\cong O_u\times \tilde{G}_u$ and $\tilde{L}_u$ are lagrangians in $T^*\operatorname{SL}_2(\mathbb{C})$.
For $u\neq 0$, $\pi_1(\tilde{L}_u)\cong\pi_1(\tilde{G}_u)$, so we could also view $G_u$ as moduli space of flat $\mathbb{C}^*$-connections on $\tilde{L}_u$.
What we want to know is if it is possible to identify $G_0$ as a moduli space of certain types of connections or other objects on $\tilde{L}_0$ so that the identification should be compatible with $G_u\cong \mathrm{Hom}(\pi_1(\tilde{L}_u),\mathbb{C}^*)$, where compatibility implies a 'continuous' variation. Since $\pi_1(L_0)$ is finite, flat connections cannot be considered, but there may exist families of algebraic D-modules that satisfy these conditions.
Our attempt involved a version without Lagrangians. Let $\tilde{X}$ be the universal centralizer of $\operatorname{SL}_2\mathbb{C}$ defined similarly to that of $\operatorname{PGL}_2\mathbb{C}$.
Then $\tilde{X}=\{(a,b,u)\in\mathbb{C}^3\mid a^2-b^2u-1=0\}$. For $u\neq 0$, the corresponding $D$-modules take the form $$\frac{\mathbb{C}\langle a,b,\partial_a,\partial_b\rangle}{(a^{2} -b^{2} u -1,a\partial_b-u b\partial_a,\frac{a}{\sqrt{u}}\frac{\partial }{\partial b}+b\sqrt{u}\frac{\partial}{\partial a}-\alpha)}$$ These $D$-modules correspond to the points $\begin{bmatrix} \frac{q+1}{2} & \frac{q-1}{2}\sqrt{u}\\ \frac{q-1}{2\sqrt{u}}& \frac{q+1}{2} \end{bmatrix}$ in $\operatorname{PGL}_2\mathbb{C}$, where $q=e^{2\pi i \alpha}$ represents the holonomy. In order for such D-module to have limit as $u$ approaches $0$, it will lead to the form $\alpha=\frac{\beta}{\sqrt{u}}$ for some constant $\beta$. The limit is then given by
$$\frac{\mathbb{C}\langle a,b,\partial_b\rangle}{(a^{2} -1,a\frac{\partial }{\partial b}-\beta)}$$
This is indeed a family of $D$-modules parametrized by $\beta\in\mathbb{C}$, and can be characterized as isomorphism classes of invariant $D$-modules of rank 1 on $\tilde{G}_0$. However, we do not have a nice method to identify these $D$-modules with the points of $G_0$, as $\lim_{u\to 0}\begin{bmatrix} \frac{q+1}{2} & \frac{q-1}{2}\sqrt{u}\\ \frac{q-1}{2\sqrt{u}}& \frac{q+1}{2} \end{bmatrix}$ does not exist.
