The moduli space of Higgs bundles over a curve is a well known (singular) variety with an extremely rich geometry, in particular it is hyperKähler and becomes an integrable system after being equipped with the so called Hitchin morphism which, to any Higgs bundle, associates a finite cover of the base curve named spectral curve. Associated to the hyperKähler structure, Kapustin and Witten introduced in 2007, BBB and BAA-branes, predicting that they occur in pairs dual under mirror symmetry. An example of BBB-brane is a hyperKähler bundle supported on hyperKähler subvariety, and an example of BAA-brane is a flat bundle supported on a complex Lagrangian subvariety. Hitchin described in 2019 a family of subintegral systems lying on the critical loci of the Hitchin integrable system parametrized by spectral curves with a fixed number of singularities. The critical subsystem obtained by considering spectral curves with maximal number of singularities is a hyperKähler subvariety and the author, along with Oliveira, Peón-Nieto and Gothen, studied the BBB-branes constructed over it, and their image under Fourier-Mukai transform, which are supported on complex Lagrangian subvarieties. Surprinsingly, Hitchin showed that the critical subsystem obtained by considering spectral curves with 1 singularity is not a hyperKähler subvariety and he conjectured that only the critical subsystem with a maximal number of singularities is hyperKähler.

In this work, joint with Hanson, Horn and Oliveira, we study the critical subsystems with any number of singularities, showing that their image under Fourier-Mukai is supported on a certain family of complex Lagrangian subvarieties which we describe.