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We derive a new formula for the Darboux matrix under the assumption that the square of the Darboux matrix vanishes for some values of the spectral parameter. The motivation and main application for this approach is a recently introduced class of spectral problems in Clifford algebras. As an example, we consider isothermic surfaces.
We present a class of spectral problems in Clifford algebras, linear in the spectral parameter. They are associated with integrable systems which describe various types of orthogonal coordinates in Euclidean spaces and some other orhogonal nets. In particular, we discuss isothermic surfaces with arbitrary codimension and a generalization of the sine-Gordon equation.
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