Canonical explicit Bäcklund transformations with spectrality for constrained flows of soliton hierarchies

It is shown that explicit Bäcklund transformations (BTs) for the high-order constrained flows of soliton hierarchy can be constructed via their Darboux transformations and Lax representation, and these BTs are canonical transformations including Bäcklund parameter η and possess a spectrality property with respect to η and the ’conjugated’ variable μ for which the pair (η, μ) lies on the spectral curve. As model we present the canonical explicit BTs with the spectrality for high-order constrained flows of the Kaup-Newell hierarchy and the KdV hierarchy.