  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 2 vector bundle. We
define a Hitchin map and give a spectral correspondence. We also state a
Hitchin-Kobayashi correspondence for a generalization of the Hitchin 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.
