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42) (with i = 0), we obtain still another formula: E{x,y\k) = -(2n)~nkn~2 z I n e{x^\-k)e{y^\k)dw. 2)). 1). We also note that E(x,y]-k) = (-l)nE(x,y;k). 4) Next introduce the operator E+(k) : B(Rn) for any k E C + \ { 0 } by —> S * ( R n ) , defined £ + ( * ) = \(2nrk»-*T+(k)*JT+(k). 5) Here T+(&) : B(Rn) —> Χ 2 ( 5 η _ 1 ) is the trace operator introduced in Section 4, and J is the unitary operator in L 2 ( S n _ 1 ) defined by J : φ(ω) —► φ(—ω). 1. The operator E+(k) (any k G C + \ { 0 } ; . (Rn), E+(k)f= f n has the following eik^E(x,y;k)eik^f(y)dy.

8] Adler, M. : T h e Kowalewski and Hénon- Heiles Motions as Manakov Geodesic Flows on SO(4) - a Two Dimensional Family of Lax Pairs, Commun. M a t h . Phys. 1 1 3 , pp. 659-700(1988). [9] Adler, M. : T h e Complex Geometry of the Kowalewski-Painlevé Analysis, to appear in Inventiones M a t h . 1989, preprint May 1988. [10] Arnold, V. : Mathematical Methods of Classical Springer, New York - Heidelberg - Berlin (1978). : A Singularity Analysis of Integrability and Chaos in Dynamical Systems, in Singularities and Dynamical Systems, Proceedings, Heraklion, Greece, North-Holland-Amsterdam (1983).

Incidentally, the quadratic differential equations for yi/ya provide a useful method for computing their Taylor series, with initial conditions on D[m~ or D2~ . Therefore the coordinates yi/yQ can be expressed as Taylor series in tf, with holomorphic coefficients. 8)) lead~ ). The ing to such ratios denoted by z ( £ , p , D \ m ) and z(t,p,D2 latter are the meromorphic extensions of the original Painlevé solutions z ( i , p , D\m~~ ) and z(t,p, D2~ ), as discussed in the coherence condition. ( m ~ 2 ) yields the special Painlevé solution z(t,p,D(m~2>) PAINLEVE SOLUTIONS 31 promised earlier.

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