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

a Hamiltonian flow which interpolates the billiard motion at integer times t € 4/Z. The

novel feature here, as in all cases where m ^ iV, is that the loop A(\) and hence the

loop e*logA(A), is not invertible at A = 0 6 S = iR. Thus the factorization problem is

nonstandard in a fundamentally new way.

(2) The above flow induces a flow on the unit sphere bundle Y = {(x,y) : (x,C~2x) =

1, ||y|| = 1} to the ellipse, which is also Hamiltonian in the symplectic structure obtained

by restricting the standard two form 2 i = i ^x* ^ dyi on R2N, to the submanifold Y. The

induced flow is completely integrable.

(3) In the case N = 2 the flow takes a particularly simple form,

— y®y = ,3[xAy,y® y]

dt

d

(1.45)

-xAy = f3[y®y,C2} ,

where f3 is a calculable constant. These equations can be reduced further to the pendulum

equation and then solved in the standard way in terms of an elliptic function.

In an earlier paper Moser [M] considered a class of integrable systems which give rise

to isospectral deformations of rank 2 extensions

M(x,y) =A + ax®x + bxO)y + cy®x + dy®y (1.46)

of a fixed, real symmetric matrix A. Here

A = ad-bc^Q. (1.47)

In the final section, Section 5, we show that all the examples considered by Moser can

be interpreted in terms of our loop group framework, in the following sense. Associated

to every isospectral deformation t t— • Ad(x(t),y(t) ), there is a loop ^4(x,y,A), with (1 —

A2) A(x,y, A) quadratic in A, and a Hamiltonian flow

±A(x(t),y(t),X) = [(x.F'(A(x(t),y(t),-)))M A(x(t),y(t),\)} (1.48)