We consider an differential equation $$ d_A: \frac{dW}{dz} = A(z) W$$ with $A(z) \in \mathit{Mat}(n, \mathcal{O}_{\mathbb{C^*}})$ and $W(z)$ a fundamental system of this equation. If I now consider the universal cover $p: \mathbb{C} \rightarrow \mathbb{C^*}, z \mapsto e^z$ and the deck transformation $\tau_n : \mathbb{C} \rightarrow \mathbb{C}, z \mapsto z + 2 \pi i n$, i get the analytic continuation of $W(z)$ namenly $\tilde{W}(z) = (W \circ p)(z)$. Plugging everything in, leads to $$ \begin{eqnarray} \tilde{W}(\tau_n(z)) =\tilde{W}(z + 2 \pi i n) &=& W(e^{z + 2 \pi i n}) = W(e^z)W(e^{2 \pi i n})\\ &=& \tilde{W}(z) \tilde{W}(2 \pi i n) = \tilde{W}(z) M_A(W) \end{eqnarray} $$ with $M_A(W) \in GL(n, \mathbb{C})$ the monodromy matrix.
My question now, is the follwing:
I now want to show, that both $\tilde{W}(z)$ and $\tilde{W}(z + 2 \pi i n)$ are fundamental systems, but now for an "lifted equation along p", lets say
$$ d_{\tilde{A}}: \frac{d\tilde{W}}{dz} = \tilde{A}(z) \tilde{W}$$
where $\tilde{A}(z) \in Mat(n, \mathcal{O}_{\mathbb{C}})$. My only idea, was to plug in the definition of $\tilde{W}$ and use that $W$ is an solution to $d_A$:
$$\frac{d\tilde{W}}{dz} = \frac{d(W \circ p)}{dz} = \frac{dW}{dp} \frac{dp}{dz} = A(p(z))W(p(z)) \frac{dp}{dz}=\tilde{A}(z) \tilde{W}$$
where $\tilde{A}(z) = \frac{A(p(z))dp}{dz}$. My problem is, that i dont know if this is right, especially the expression i get for $\tilde{A}$. And also, this does not work very well for $\tilde{W}(z + 2 \pi i n)$. Im thankful for help and hints :)