In this paper we study four families of moduli problems, which give rise to two dimensional examples of the Hitchin map. The spectral curves arising as fibers of the Hitchin map are described explicitly in terms of linear series. Careful analysis is done to show that the spectral curve may be replaced by its reduced subscheme; said another way, the minimal polynomial of the Higgs field is always the radical of the characteristic polynomial. Certain higher rank bundles are considered, and a result akin to Atiyah's result for vector bundles on an elliptic curve shows that the higher rank problems are equivalent to the lowest rank case in each family.