(2.122)
36 P. DEIFT, L. C. LI, AND C. TOMEI
where M{i) is real and skew.
Finally, to prove (2.117), it is sufficiently the group property for flows, to verify
(2.117) for k = 1. But for k = 1,
log(M0(A)) _ /  AMp  A 2 J 2
1  A2
= ( 4 ^ ) ( T ^ )
by(1

20)
But these factors satisfy (1.29), (1.30) and so
«7_(l,A) = ( u ;
0
 A J ) / ( l  A )
g+(l,X) = (^ + XJ)/(l + X)
by uniqueness. Thus
j  i  j ^ = M(l,X) = g+(l,X) x ( j—^5 )g+(l,X)
_
.W'Q
+ XJ.  i
,WQ
+ XJ. ,u0 — AJs ,u£ + XJ
~ \ 1 + X ' ( l + X )( 1  A ^ 1 + A
Aft (A)
1  A 2
_ J  A M !  A 2 J 2
1  A 2
which completes the proof. D
The Hamiltonian H generates (2.115) on the coadjoint orbit OA" where
^ ° = ( 1  A M
0
 A 2 J 2 ) / ( 1  A 2 )
and
A? = (J 2  Mo  l)/4 , A°0 = (1 + J 2 )/ 2 and A ^ = (J 2 + M
0
+ l)/ 4 (2.123)
(see Section 2(b)). By the results of the previous subsection OAQ is generic and of dimension
4N2 — 4N provided A J (and hence A!L\ — {A\)T) is invertible and has distinct eigenvalues
(given J, this is clearly true for a dense open set of Afo's). The jTrfc's and the Arfc's of