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10) for any / € L^(Rn). In the sequel we shall use the following notation for strips in the complex plane. Σα = {k eC: |Imfc| < a}, Σ ± = {fc 0 ) + t " 1 Z ( 0 '°'^ ) + O(1) · They have eight special Painlevé solutions, depending on 4 parameters A,B,C andD. / ~ . x nf A A iej,6a) —δα ieA\ + 0(t2) _ N 7 = *"2(1,0,te) + i° ( j , —-, — J + 0(t). A second set corresponding to the 4 points (e, 77 = ± 1 ) where the flow is doubly tangent to Όη (osculating solutions) behaves as: m = βηΖ . VW + C 2 7 = ( -t~ , t~3, . € yJWTC - | ^ ~ 2 , -2e\/D2 t^^mt"1 ) + higher terms J + C J + higher order terms. For generic A,B,C,D, the generic and the eight special Painlevé solutions above are the only asymptotic solutions to the Kowalewski system. Hence T+(k) is surjective. 3 (without precise constants) follows from the results of [4]. 3 for k real was given by Strichartz [13]. P r o o f of T h e o r e m _ 4 . 3 . (ii)=^ (i). 12) with φ £ tfiS"-1) k G C + \ { 0 } . 1). 10): = (r+(jfc)»(x). eik^u(x) Since Ran(T+(&)*) C £ * ( R n ) , it follows that eik^u e £ * ( R n ) as claimed. (i) => (ii). 1) such that elk\x\u £ i ? * ( R n ) , k G C + \ { 0 } . 17) ur = π · V u = -|^. 18) for all R > 1. 18) implies there exists a sequence { r j } , j = 1 , 2 , .