The moduli space of G-local systems on a surface is a fundamental object in mathematics. The cases $G=SU(2)$ and $G=SL_2(\mathbb{R})$ are of particular interest. Consider the group $E$ of isometries of the Euclidean planes. It is much simpler than two previous groups and is a semi-direct product.
Question: What is known the moduli space of $E$-local systems on a surface? It should be in some sense a «curvature 0» limit of the $SU(2)$ case. What facts about $SU(2)-$local systems pass to the Euclidean case?
One might expect that Weyl-Peterson form will lead to a contact structure and that moduli space of $E$-local systems on 3-manifold, bounding a surface, gives a Legendrian in it.
