4 P. DEIFT, L. C. LI, AND C. TOMEI

(see [V]),

Mfc+i =u;kMkujZ , (1.12)

where

Mk = u%J - Jw& £ o(N) . (1.13)

In the variables (a;*, M*), the nature of the above correspondence is refelcted in the fact

that u*k is not uniquely determined by Mk through (1.13). The choice of a particular

mapping \£ reduces, given Mk, to a particular choice of matrix uk £ 0(N) in (1.13).

Moser and Veselov proceed as follows. They consider the closed 2-form

u = tr dXJ A dYT = ^ J, dXij A d*-,- (1.14)

restricted to Q2n = O(iV) x O(N). A straightforward, but somewhat tedious, computation

shows that,

oj is nondegenerate at (X,Y) £ O(N) x O(N)

& (1.15)

A + A' ^ 0 for all A, A' £ s p e c ( r T X J " 1 ) .

In other words, a; is nondegenerate at a point (Xk-i,Xk) £ Q2n only if A -f A' ^ 0

for all (generalized) eigenvalues A, A', det(ujt — A J ) = det(u;jfc — A'J) = 0, where again

ujk = X^Xk-i- The basic observation in [MV] is that (1.13) is equivalent to the matrix

polynomial factorization

Mk(\) = I - AM* - A2 J 2 = (uJ + \J)(«k - A J ) (1.16)

and switching factors

M*+1(A) = (u;*-AJ)(u;jf + AJ)

(1.17)

=

I-\Mk+i-\2J2

yields Mk+i = u;fcMfcu;jf, by (1.12). For (Xk-i^Xk) satisfying (1.15), the above factoriza-

tion for Mfc(A) has the property that for

S = {A : det( J - AAfjb - A2 J 2 ) = 0} ,

S+ = {A : det(u;* - AJ) = 0} , (1.18)

S- = {A:det(wjf+ AJ) = 0} ,