Rank-2 wobbly bundles from special divisors on spectral curves

We study rank-2 wobbly bundles on a Riemann surface $C$ of genus $g\geq 2$, i.e. semi-stable bundles admitting nonzero nilpotent Higgs fields, in terms of direct images of line bundles on smooth spectral curves $\tilde{C} \overset{\pi}{\rightarrow} C$. We give a sufficient condition for a semi-stable bundle $E$ to be wobbly: $E$ is a twist of $\pi_\ast \left(\mathcal{O}_{\tilde{C}}(\tilde{D}) \right)$ where the norm of $\tilde{D}$ is a summand of the divisor of a quadratic differential on $C$. We sketch the proof of the necessary condition statement, namely all rank-2 wobbly bundles can be characterised as such, and discuss how certain singularities of the wobbly locus arise from the Brill-Noether loci of spectral curves.