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By L. Auslander, R. Tolimieri

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48) 54 P. D E I F T , L. C. LI, AND C. 45) can be solved by standard techniques, and we sketch the approach for the convenience of the reader (see [G-K]). 49) (gt)-1=I+h. 50) where /i_ and h+ + (I — g&) belong to £ 2 ( E ) . 50) can be rewritten as (W+g*/+ for suitable fi±,v± G 7r±(L2(Yl)). 51) is a bijection: on the other hand, n±g^'K± — ^±{g^ — I)n^ are compact on L 2 (E) by standard computations. Hence the diagonal operator 7r+# # 7T+ 0 0 7r_(7#7r_ is Fredholm of index zero. 53) has a unique solution fi- in 7r_Z,2(E) for any z/_ 6 T T _ I 2 ( E ) .

A°(X): + zoo) = A°(XJ — zoo), we must have — zoo) = 2m d i a g ( & i , . . j) Now for A € E. B(X) = (logA°)*( —A) is a logarithm of indeed e B ( A ) = e^ogA°)m(~X) = A°(-\)* = A°(A), and as B(XJ + zoo) is also real, we must have logA 0 (A) = ( l o g . 4 ° r ( - A ) . 33) 52 P. D E I F T , L. C. LI, AND C. T O M E I This implies \ogA°(xj + zoo) - log A0(xj = log A°(xj - zoo) + log A°(a:_j + 200) - \ogA°(x-j + zoo) - log A°(xj - zoo) + (log A°)*(XJ - 200) + 200) - (log A°)*(XJ - zoo) = 27rz diag(&i,...

But this follows trivially from the invariance of the determinant. 81), dIrk(A)(z = 1) = 0 . 85) C £* n g . 84). Thus the entries of Ap provide additional integrals for the Irk flows. On generic orbits 0A most of the /rjfc's degenerate: we compute the nontrivial intgrals that remain. For (z - l)A(z) = Ap + (z - l ) ( . 87) LOOP G R O U P S , INTEGRABLE SYSTEMS, AND RANK 2 EXTENSIONS 29 where Er(-) is the elementary symmetric function of degree r in the entries of M , Er(tM) trEr(M) = for all t G R.

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