  In this paper we investigate the space of harmonic maps from a 2-torus to
$\mathbb{S}^3$ using the spectral curve correspondence and Whitham
deformations. In an open and dense subset of a parameter space we find that the
space of harmonic maps is smooth and has dimension two. We also show that the
points that correspond to minimal tori (conformal harmonic maps) are either
smooth points of dimension two or singular.
