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].
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