LOOP GROUPS, INTEGRABLE SYSTEMS, AND RANK 2 EXTENSIONS 5
we have
5 = 5
+
U 5 _ , 5 ± = S ± , S+ = -S- and S+ n 5_ = 0 . (1.19)
Conversely, using a technique which is closely related to the solution of a well-known matrix
Riccati equation arising in control theory (see, for example, [S-H], [BG-M]), the authors
show that given a quadratic pencil J AM A2 J 2 , for which the associated spectrum S
has a splitting satisfying (1.19), then there exists a unique factorization
/ _ AM - A2 J 2 =
(OJT
+ XJ)(u - \J)
with 5+ = {A : det(u;-AJ) = 0} and 5_ = {A : det(uT+AJ) = 0}. This leads the authors
to the following procedure for the solution of the Euler-Lagrange equations, and hence to
a particular branch # of the correspondence determined by (1.4): given (X_i,.Xo) £ Q2n
satisfying (1.15), set
M0(A) = / - AM0 - A2 J 2 = (u # + \J)(LO0 - A J ) , a0 = XfX-x . (1.20)
Exchanging factors
Afi(A) = I - AMa - A2 J 2 = (u;0 - AJ)(u;J + XJ) , (1.21)
is an isospectral-action,
det Mi(A) = det M0(A)
and hence 5(Mi(A)) = S(M0(\)) has a splitting
5±(M
1
(A)) = 5
±
(Mo(A))
which clearly satisfies (1.19). It follows that Mi (A) has a (unique) factorization
Mi(A) = (u ^ + A./)(w! - A J ) ,
W l
G O(N) (1.22)
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