Let $X$ be a smooth curve over an algebraically closed field of characteristic 0. The category of quasi-coherent $D$-modules $\mathcal{D}_X$-$Mod$ is a symmetric monoidal abelian category. We can therefore consider algebras internal to it.
Definition: A $\mathcal{D}$-algebra over $X$ is an algebra object in the category $\mathcal{D}_X$-$Mod$.
In the smooth affine case $X=SpecA$ this is equivalent to an $A$-algebra $R$ equipped with a derivation restricting a non-trivial derivation $A$ (up to gauge equivalence). In other words a horizontal subbundle of the tangent bundle of $R$. So this is clearly some kind of generalization/analogue of a connection on a jet manifold. Since $X$ is a curve the subbundle is 1-dimensional and so is closed under lie bracket so the connection is forced to be flat.
Definition: Let $G$ be a smooth algebraic group over $k$. A principal $G$-bundle with (flat) connection over $X$ is a $\mathcal{D}_X$-algebra $R$ whose underlying scheme is a $G$-torsor over $X$ in such a way that the action of $G$ is by $\mathcal{D}_X$-algebra homomorphisms.
Definition: A flat vector bundle over a principal $G$-bundle $P=SpecR$ with flat connection is a module over $R$ internal to $\mathcal{D}_X$-$Mod$ whose underlying sheaf on $P$ is locally free finite rank.
We can now roughly phrase the fundamental theorem of Picard Vessiot theorem as follows (?):
Picard-Vessiot: There's a fully faithful functor from the category of $\mathcal{D}_{k(X)}$-modules (modules over the generic fiber of $\mathcal{D}_X$) with isomorphisms to a category whose objects are pairs $(P,M)$ where $P$ is a flat $G$-bundle over $k(X)$ and $M$ is a $G$-equivariant flat vector bundle over $P$.
Morally this seems to suggest that any $\mathcal{D}$-module over a curve has (generically at least) a "resolving covering" - a flat $G$-bundle which lifts the $D$-module to a flat vector bundle.
Is there a global version which goes along the lines of:
Very Big Dream: There's a fully faithful functor from the category of coherent $\mathcal{D}_{X}$-modules with isomorphisms to a category whose objects are pairs $(P,M)$ where $P$ is a flat $\mathcal{G}$-bundle (with $\mathcal{G}$ now an affine algebraic group scheme over $X$) over $X$ and $M$ is a $\mathcal{G}$-equivariant flat vector bundle over $P$.
