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This paper is a survey of our recent results on the bispectral
problem. We describe a new method for constructing bispectral algebras
of any rank and illustrate the method by a series of new examples as well
as by all previously known ones. Next we exhibit a close connection of
the bispectral problem to the representation theory of W1+∞–algerba. This
connection allows us to explain and generalise to any rank the result of Magri
and Zubelli on the symmetries of the manifold of the bispectral operators...
We define Bäcklund–Darboux transformations in Sato’s Grassmannian.
They can be regarded as Darboux transformations on maximal algebras
of commuting ordinary differential operators. We describe the action of these
transformations on related objects: wave functions, tau-functions and spectral
algebras.
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