Remarks on Higgs bundles twisted by a vector bundle

For any [Formula: see text]-twisted Higgs bundle on a compact connected Riemann surface [Formula: see text], where [Formula: see text] is a holomorphic vector bundle of rank two over [Formula: see text], there are two associated Higgs bundles on [Formula: see text], twisted by line bundles, which are constructed using a Hecke transformation on [Formula: see text]. We characterize all such pairs of Higgs bundles (twisted by line bundles) given by the [Formula: see text]-twisted Higgs bundles. Using this characterization, we provide a spectral correspondence for the moduli space, identifying [Formula: see text]-twisted Higgs bundles with the direct images of certain rank one torsionfree Higgs sheaves twisted by a line bundle on a spectral covering of the curve [Formula: see text].