$i=1,2$. Then $\varphi_{c'}^*(G_1(\log\,D),\nabla_1) \cong (G_2(\log\,D),\nabla_2)$, where $\varphi_{c'}:{{\mathds C}}\times T\rightarrow {{\mathds C}}\times T$ is defined as $\varphi(\theta,t)=(\theta,c'\cdot t)$ for some $c'\in{{\mathds C}}^*$.

By definition, the complement of $D^*$ in $V^*$ is an (open) orbit of the dual action of $G$, hence, there is $g\in G$ with $g(f_1)=f_2$. Then $g(h)=\chi_h(g)\cdot h$ and it follows that $\varphi_{\chi_h(g)}^*(G_1(\log\,D),\nabla_1) \cong (G_2(\log\,D),\nabla_2)$.

In order to relate the above objects to the Bernstein polynomial of $h$, we recall how the Gauß-Manin system $G(*D)$, seen as a left ${{\mathds C}}[\theta,t]\langle\partial_\theta,\partial_t\rangle$-module is obtained as a direct image of a differential system on $V$. A similar reasoning as in the next lemma can be found in [@DS proposition 2.7]. We consider $f$ as a morphism $f:V\rightarrow R=\operatorname{\textup{Spec}\,}{{\mathds C}}[r]$, and put $\Phi:=(f,h):V\rightarrow R\times T$.

\[lem:CompDirectImage\] Let $\Phi_+ {\mathcal{O}}_V(*D)$ be the (algebraic) direct image complex of the holonomic ${\mathcal{D}}_V$-module ${\mathcal{O}}_V(*D)$. Then

1.  The cohomology sheaves of $\Phi_+{\mathcal{O}}_V(*D)$ are ${\mathcal{D}}_{R \times T}$-coherent, holonomic and regular.

2.  $\Phi_+{\mathcal{O}}_V(*D)$, seen in ${\mathcal{D}}^b({\mathcal{D}}_{R\times T /T})$ is represented by $$\left(\Phi_*\Omega^{\bullet+n-1}_{V/T}(*D)[\partial_r],d-(df\wedge-\otimes \partial_r)\right).$$ Under the isomorphism $$\label{eq:GM-DirectImage}
    {\mathcal{H}}^0(\Phi_+{\mathcal{O}}(*D)) \cong \frac{\Phi_*\Omega^{n-1}_{V/T}(*D)[\partial_r]}{(d-df\wedge-\otimes \partial_r)\Phi_*\Omega^{n-2}_{V/T}(*D)[\partial_r]}$$ the action of $\partial_t$ on a class $[\omega]\in {\mathcal{H}}^0(\Phi_+{\mathcal{O}}_V(*D))$ represented by a form $\omega\in {\mathcal{H}}^{n-1}_{V/T}(*D)$ is given by $$\label{eq:ConnectionParameter-BeforeFL}
    \partial_t([\omega]) := \frac{1}{n\cdot t}\left([\operatorname{\textup{Lie}}_E(\omega)]-[\operatorname{\textup{Lie}}_E(f)\omega\otimes\partial_r]\right)$$

3.  Put $M:={{\mathbb H}}^0(R\times T, 