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 S± 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)