  We investigate 4D Chern-Simons theory with ADE gauge symmetries in the
presence of interacting Wilson and 't Hooft line defects. We analyse the
intrinsic properties of these lines' coupling and explicate the building of
oscillator-type Lax matrices verifying the RLL integrability equation. We
propose gauge quiver diagrams Q$_{G}^{\mu }$ encoding the topological data
carried by the Lax operators and give several examples where Darboux
coordinates are interpreted in terms of topological bi-fundamental matter. We
exploit this graphical description $\left( i\right) $ to give new results
regarding solutions in representations beyond the fundamentals of $sl_{N}$, $%
so_{2N}$ and $e_{6,7}$, and $\left( ii\right) $ to classify the Lax operators
for simply laced symmetries in a unified E$_{7}$ CS theory. For quick access, a
summary list of the leading topological quivers Q$% _{ADE}^{\mu }$ is given in
the conclusion section [Figures 29.(a-e), 30.(a-d) and 31.(a-d)].
