Rank-2 wobbly bundles from special divisors on spectral curves

Let X be a compact connected Riemann surface of genus g ≥ 3 and S → X a $2:1$ branched covering defined by a generic quadratic differential on X. In the following notes, we explore how taking direct images of line bundles defined by certain distinguished special divisors on S gives rank-2 bundles having nilpotent Higgs fields on X. The key ingredient is the notion of Baker-Akhiezer divisors associated to Higgs bundles and choices of subbundles of the underlying rank-2 bundles.