By H. J Chaytor

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Inserting eq. 16) into eq. 10) one gets Mk = P (k) (L, λ) . More− over, in this formula the Laurent expansions of the coeﬃcients of P (k) can be truncated at some positive power of λ − λk since a high enough power cannot contribute to the singular part, yielding a polynomial with coeﬃcients Laurent polynomials in λ − λk . It is important to realize that the dynamical variables are the matrix elements of the Lax matrix, or the matrix elements of the Lk,r . 2 The Zakharov–Shabat construction the number and the order of the poles of the Lax matrix amounts to specifying a particular model.

The reduced equations of motion are equivalent to the equations of motion for the Neumann model. Indeed, the reduced system is characterized by the conditions t XX = 1 and t XY = 0, but the equations of motion eq. 23) do not preserve the second condition. We need to perform simultaneously a time-dependent gauge transformation Y → Y +λ(t)X to keep the motion on the gauge surface. Writing: 0= d ˙ − λJX) (X, Y + λX) = (−JX, Y + λX) + (X, −JY − L0 X + λX dt = −(X, L0 X) + λ˙ since J is antisymmetric, gives λ˙ = (X, L0 X).

Note that this is a two-poles r-matrix with poles at λ = ±µ. Example 2. We consider next the Neumann model. Recall that the Lax matrix reads L(λ) = L0 + λ1 J − λ12 K. As in the Euler top it satisﬁes t L(−λ) = L(λ). Hence we are dealing with the graded orthogonal group t g −1 (−λ) = g(λ). Let us check that the matrix L (λ) is an orbit under − the coadjoint action. As a matter of fact: (g(λ)L− (λ)g −1 (λ))− = − 1 1 g0 K t g 0 + (g0 J t g 0 − g1 K t g 0 + g0 K t g 1 ) 2 λ λ with g(λ) = g0 +λg1 +. ..