LOOP GROUPS, INTEGRABLE SYSTEMS, AND RANK 2 EXTENSIONS 9

(3) as anticipated from [MV], the flow t H- M(t,\) = {I - XM(t) - A2 J 2 ) / ( l - A2)

induces an integrable Hamiltonian flow t »-* M(t) on the dual Lie-algebra o(N)* equipped

with the standard Lie-Poisson structure.

In Section 3 we consider the general case where (1.24) may fail and 5 = 5+ U 5_

is any decomposition satisfying (1.19). As we show, this leads to a loop algebra, and a

Riemann-Hilbert factorization problem, on a skeleton S = Uj=-7 ^ i composed of lines S j

(asymptotically) parallel to iR with So = iR and Ej = —E_j. All the results of Section 2

extend to this situation, but we focus on the novel aspects of the Riemann-Hilbert problem.

The authors in [MV] also consider the more general problem S = Ejt tr XicJX'[+v

where Xk now lies in the Stiefel manifold Vm%iw of in x N matrices satisfying XkX^ = Im.

This leads to a factorization problem for quadratic matrix polynomials of the form

A2 J 2 + AM, - XLiXk-i . (1.42)

In Section 4, we concentrate on the case where m = 1, leaving the more general problem,

with its many connections to the work of Adams, Hamad and Previato [AHP] and Schilling

[S], to a later publication. We focus in particular on the billiard ball problem in an elliptic

region E = {x :

(x,C~2x)

1} in J R ^ , where C is positive and diagonal. Although the

variational functional S = Ylk \Xk "~ ^fc-il* (xk*C~2Xk) = 1, is not of the above type,

Moser and Veselov show that the solution of the billiard ball problem in E also reduces to

a polynomial matrix factorization problem for a quadratic polynomial

1(A) = y ® y + A a r A y - A2C2 , (1.43)

where

(x,C" 2 x ) = l and (y,y) = l . (1.44)

Our results in Section 4 are the following.

(1) As before, the coadjoint orbit of G through A{\) = L(X)/(1 - A2) provides a sym-

plectic structure for the problem and the analog of the factorization (1.3S) again leads to