I originally posted the question on math.stackexhange, but there doesn't seem to be an answer. I apalogize in advance for cross posting.

Let $E\rightarrow X$ be a holomorphic vector bundle over a compact Riemann surface with a holomorphic connection $\nabla:E\rightarrow E\otimes K$, where $K$ is the canonical bundle of $X$. Since the holomorphic connection is necessarily flat, its sheaf of local holomorphic sections $\mathcal{E}$ defines a (holonomic) $D$-module. Every holonomic D-module is locally cyclic, i.e. for any point $z_0$ there exists a neighborhood $U$ s.th. $\mathcal{E}(U)$ has a cyclic generator as a $D$-module (see e.g. Proposition 3.1.5. in Björk: Analytic $D$-Modules and Applications). Suppose we are given a coordinate $z$ on $U$ and identify $D(U)\cong D_1$, with $D_1=\mathbb{C}\left\lbrace z \right\rbrace \left\langle \partial_z \right\rangle$ (differential operators with coefficients in convergent power-series). So locally it holds $\mathcal{E}(U)\cong D_1/ I$, where $I$ is the ideal of differential operators annihilating the cyclic generator. This ideal is in general generated by two elements $P,Q$, with $P$ an operator of smallest possible degree in $I$ and furthermore $I/D_1P$ is of torsion type, i.e. for any $D\in I$ it holds $z^nD\in D_1P$ for some $n$ (Proposition 5.1.4 and Remark 5.1.5 in Björk: Analytic $D$-Modules and Applications). This implies for the dual $D$-module $\hom_{D_1}(D_1/I,\mathbb{C}\left\lbrace z\right\rbrace)=\left\lbrace f\in \mathbb{C}\left\lbrace z\right\rbrace \, \middle|\, Pf=Qf=0\right\rbrace=\left\lbrace f\in \mathbb{C}\left\lbrace z\right\rbrace \, \middle|\, Pf=0\right\rbrace$.

So far so good. Now on $U$ the holomorphic connection reads $\nabla|_U=\partial+A$ with $A$ some matrix of holomorphic functions.The dual bundle naturally comes with a holomorphic connection, too, which in local coordinates takes the form $\partial-A^T$. The whole discussion above shows that locally flat sections ($(\partial-A^T)Y=0$) are in one to one correspondence with solutions of $Pf=0$.

On the other hand there is Deligne's lemma of a cyclic vector. One way to formulate it, is to say that locally on a coordinate neighborhood $U$, for a vector bundle with holomorphic connection there exists $G\in \mathrm{GL}(n,\mathcal{O}(U))$, s.th. \begin{equation} \partial_z G-G A^T=\tilde{A}G \end{equation} with $\tilde{A}$ in companion form. Here $\partial-A^T$ is the local form of the holomorphic connection. But in general the non unit entries $a_i$ in $\tilde{A}$ are only meromorphic and $G$ might not be invertible as a holomorphic matrix.

It is clear that a system of linear differential equations $\partial_z Y=A^T Y$ with $A^T$ in companion form corresponds to a single $n$-th order scalar differential equation $Qf=0$. So from Deligne's cyclic vector lemma I get an $n$-th order scalar differential equation, but the corresponding differential operator might not be in $D_1$, but in $\mathbb{C}\left\lbrace z\right\rbrace [z^{-1}]\left\langle \partial_z\right\rangle$.

**Q**: Is there any relation between the differential operator I get from the discussion in the first paragraph applied to the dual bundle and the differential equation I get from Deligne's cyclic vector lemma?

I guess they are the same, maybe after imposing further constraints on the open $U$. It might very well be that the relation is obvious and just shows my lack of understanding.