In this paper, we consider a generalization of the theory of Higgs bundles over a smooth complex projective curve in which the twisting of the Higgs field by the canonical bundle of the curve is replaced by a rank [Formula: see text] vector bundle. We define a Hitchin map and give a spectral correspondence. We also state a Hitchin–Kobayashi correspondence for a generalization of Hitchin’s equations to this situation. In a certain sense, this theory lies halfway between the theories of Higgs bundles on a curve and on a higher-dimensional variety.