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  1. We use the companion matrix construction for [Formula: see text] to build canonical sections of the Chevalley map [Formula: see text] for classical groups [Formula: see text] as well as the group [Formula: see text]. To do so, we construct canonical tensors on the associated spectral covers. As an application, we make explicit lattice descriptions of affine Springer fibers and Hitchin fibers for classical groups and [Formula: see text]. 
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