Twistorial constructions of higher genus integrability

I will present a new method to engineer integrable models in 4d with higher genus spectral parameters. The method has a twistorial origin - by working on a branched covering of twistor space, I show how one can derive deformations of holomorphic BF theory on twistor space which descend to elliptic and hyperelliptic models on R^4 via the Penrose transform. I show how one can bootstrap the Penrose transformed actions using symmetry and integrability to find deformations of self-dual Yang-Mills theory. I will also discuss some novel deformations of a BF type description of Hitchin’s equations. This is based on my recent paper: 2509.12486