Basic Representations of Genus Zero Nonabelian Hodge Spaces

In some previous work, we defined an invariant of genus zero nonabelian Hodge spaces taking the form of a diagram. Here, enriching the diagram by fission data to obtain a refined invariant, the enriched tree, including a partition of the core diagram into $k$ subsets, we show that this invariant contains sufficient information to reconstruct $k+1$ different classes of admissible deformations of wild Riemann surfaces, that are all representations of one single nonabelian Hodge space, so that the isomonodromy systems defined by these representations are expected to be isomorphic. This partially generalises to the case of arbitrary singularity data the picture of the simply-laced case featuring a diagram with a complete $k$-partite core. We illustrate this framework by discussing different Lax representations for Painlev\'e equations.