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O o ^ i . 4) (note that we always have sgn ftm+i = 1, sgn Q _ m = — 1) and define a decomposition of 42 LOOP G R O U P S , INTEGRABLE SYSTEMS, AND RANK 2 EXTENSIONS

A) = / + ( A ) , with f± G T T ± £ 2 ( S ) . As (det g)(-X) = AGS, det g(X) and as (det g)(X) > 0 for A G S 0 , the above "Vanishing Lemma" argument now applies and we conclude that det(I+fi±(X)) = 0, which is a contradiction. Hence I + JJL± are invertible. This completes the proof of our results, which we summarize in the following theorem: T h e o r e m 3 . 5 8 . 4°(A) g+(t,X)= = t f + ( t A ) ?

110) above. ))(A) ,M(t,A)] , M(0,A) = Afo(A) generated by the Ad* -invariant Hamiltonian H(A) = ]im [^ ti{A(\)logA(\)-A(\))^f rtoo J_ir on g*. 116) algorithm 2Z . 118) For all times t, M(£, A) has the form where M(t) is real and skew, and M(k) = Af* for all k £ ZZ . 115) follows from jR-matrix theory and can be verified directly. One obtains the formula and one needs dff-(t,A)_ „ , . _ 1! (t, A)" = (TT_ log M ( t , •) )(A) . 102), 1 by ^~ ' dia g° nalit 12121") y> """ ^ = -(7_(l)-h7o) . 119). 114) that (1 — \2)M(t, A) has an analytic contin- uation to Re A > 0 and to Re A < 0, and grows at most quadratically as A —• oo.