  In this paper we study $G$-Higgs bundles over an elliptic curve when the
structure group $G$ is a classical complex reductive Lie group. Modifying the
notion of family, we define a new moduli problem for the classification of
semistable $G$-Higgs bundles of a given topological type over an elliptic curve
and we give an explicit description of the associated moduli space as a finite
quotient of a product of copies of the cotangent bundle of the elliptic curve.
We construct a bijective morphism from this new moduli space to the usual
moduli space of semistable $G$-Higgs bundles, proving that the former is the
normalization of the latter. We also obtain an explicit description of the
Hitchin fibration for our (new) moduli space of $G$-Higgs bundles and we study
the generic and non-generic fibres.
