  We study minimal annuli in $\mathbb{S}^2 \times \mathbb{R}$ of finite type by
relating them to harmonic maps $\mathbb{C} \to \mathbb{S}^2$ of finite type. We
rephrase an iteration by Pinkall-Sterling in terms of polynomial Killing
fields. We discuss spectral curves, spectral data and the geometry of the
isospectral set. We consider polynomial Killing fields with zeroes and the
corresponding singular spectral curves, bubbletons and simple factors. We
investigate the differentiable structure on the isospectral set of any finite
type minimal annulus. We apply the theory to a 2-parameter family of embedded
minimal annuli foliated by horizontal circles.
