  Let $X$ be a compact connected Riemann surface of genus $g \geq 3$ and $S
\rightarrow X$ 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.
