Nonlinear representation theory

Suppose we define an abstract notion of "wild nonabelian Hodge structure", consisting of a hyperkahler manifold with its various algebraic structures, such that it is a wild Hitchin moduli space for some choice of a group, an irregular curve and residue data. The choice of such a realisation of the abstract space can be viewed as a "representation" or "realisation" of the space. Study this representation theory.

Here we are viewing these moduli spaces as global/nonlinear analogues of algebraic groups (as the Riemann--Hilbert problem suggests).

For example, take one simple example (such as the $D_4$ Hitchin space, the Painleve VI space, $GL_2$ with four simple poles on $P^1$), of complex dimension $2$ and find all the possible representations of this space.