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

For M e o2(N)* set

H = HF(M) = J' tr [F(M(\)) - F(

J2)]

A

*_X

. (2.138)

A direct computation shows that

Vtf

F

(A/) = - / ( ^ ( A f C A ) ) 7 — . (2.139)

Substitution of (2.139) in (2.136) yields (2.135). In particular for F(x) = rrlogx - x, we

see that HF is the Hamiltonian for the flow induced on 02(N)* by the interpolating flow

(2.115). In particular the time-one map Mo — M ( l ) = Mi is Poisson on 02(N)*, as

discovered by Moser and Veselov [MV].

The above computations can also clearly be used to show that

Trk(M) = Irk(M(X))

forms a commuting set of integrals on c2(N)* for Hp The dimension of the generic

symplectic leaves of the Lie-Poisson structure on 02(N)* is |iV(Ar — 1) — [N/2], and one

verifies by a straightforward combinatorial argument that, in the generic case, precisely

|(|AT(Af — 1) — [N/2]) of the irfc's remain independent on the leaf. These integrals are of

course equivalent to the eigenvalues of the operator M + A J 2 , which were used by Manakov

[Man] to integrate the continuum Euler-Arnold equations (see application below).

We have proved the following result, which is the final theorem of this section.

Theore m 2.140. The interpolating flow (2.115) on 0 ( / _ A A / - A 2 J 2 ) / ( I - A 2 ) induces a

completely integrable Hamiltonian flow t — M(t) on 02(N)*. The induced flow has solution

M(t) = g+(t, 0)" 1 Mo g+(t, 0) , (2.141)

where g+(t,0) is orthogonal and is the evaluation at A = 0 of the solution g+(t,\) of the

Riemann-Hilbert factorization problem (2.110). D

As an application of the methods of the chapter we now show how to use the interpo-

lating flow to resolve a question raised by Moser and Veselov in [MV].