In this talk, I will describe some interesting geometry on the moduli space of $\lambda$-connections. First, I will explain what I mean by a $\lambda$-connection and show how this object can be abelianised to a $\lambda$-connection of rank one by extending the spectral correspondence for Higgs bundles to flat bundles. I will then describe how this abelianisation correspondence can be used to define interesting geometric structures on the moduli space of $\mathrm{SL}(2,\mathbb{C})$-$\lambda$-connections: namely, a complex hyperKähler structure as well as what looks like a completely integrable system but with fibres being abelian character varieties rather than abelian varieties. Based on joint work in progress with Tom Bridgeland and Menelaos Zikidis.