LOOP GROUPS, INTEGRABLE SYSTEMS, AND RANK 2 EXTENSIONS 11
arising from a factorization of type (1.3S) in which the logarithm is repalced by F'^),
with the property that for the appropriate parameter value.A = Ao, (1.48) reduces to
the given isospectral deformation. In particular we learn that the isospectral deformations
considered in [M] can be solved by a Symes-type factorization in an appropriate loop group.
Furthermore, the curve {(A, 77) : det(i4(#,y,A) 77) = 0} is precisely the curve used by
Moser in [M] to linearize the flow.
Remark. As noted in the Appendix, the presentation of the basic results of iJ-matrix
theory is given with the finite dimensional situation in mind. In the infinite dimensional
situation the results remain true after suitable modification. Our point of view is that
i?-matrix theory provides a useful and accurate guide to what is true in the loop group
situation, and we provide infinite dimensional versions of the proofs and the definitions as
they are needed in the text.
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