34

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

Theore m 2.113. Let

e

t l o g ( M o ( A ) )

=

^

+ (

^

A )

^ _ ^

A )

be the factorization (2.110) above. Then

M(t,X)=g+(tA)-1M0{\)g+(t,\)

= g-(t,\)Mo(\)g-(t,\)-1

solves the flow

(2.114)

^ % ^ = [0r.logAf(i,.))(A) ,M(t,A)] ,

M(0,A) = Afo(A)

generated by the Ad* -invariant Hamiltonian

H(A) = ]im [^ ti{A(\)logA(\)-A(\))^f (2.116)

rtoo J_ir 1 — Az

on g*. ,and at integer times interpolates the Moser-Veselov algorithm

M(k,\) = Mk(\)/(l-\2) , ke 2Z . (2.117)

For all times t, M(£, A) has the form

M(t, A) = (I - AM(0 - A2 J 2 ) / (1 - A2) , (2.118)

where M(t) is real and skew, and M(k) = Af* for all k £ ZZ .

Proof: Equation (2.115) follows from jR-matrix theory and can be verified directly.

One obtains the formula

and one needs

'-g.(t, A)" 1 = (TT_ log M(t, •) )(A) . (2.119)

dff-(t,A)_ „ , . _ !

dt