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By Manin Yu.I.

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8, Chap. 2 a ) . The fact t h a t the operator J£tBjdJ — ( — d)ioB,j i s Hamiltonian i s proved in Sec. 8 , Chap. I . 12. 2 b ) . We f i x r due to t h e system r e p l a c e d by dx 1 dxL = [ < L > , L] . 1ZoZ°) mod Im d. ,ZoZ°<*-i>= \{Lr) 7Z]oZ°= < Lr > ~oZ°*—l° < U >~o2°<*-'> = = < If )~oZ°*-[Lo < Lr > TZ 0 ^- 1 '! +Z°*= < U >~ = [ < U ) ~ L°*\-[Zo < 2/ >~ S i n c e t h e r e s i d u e s of t h e commutators l i e i n Imd L°i-% by Lemma 3 . $. According t o t h e d e f i n i t i o n s of S e c .

Proof of Theorem 2 . 1 7 . l a r , commute w i t h N L= X . s Z] = [f, L° — < L") ~\ . 17 commute p a i r w i s e and, i n p a r t i c u - Therefore, in the ring s But ord(L° — ord[ < Ls > ~, Z] > • < A 7 —2, i f a^ l i e s i n t h e c e n t e r of Therefore, 33 (use i s made of -1 C o r o l l a r y 2 . 3 which remains v a l i d for composition i n S((S )). by formula 3. (2)). 1. I n t h i s s e c t i o n we s e t 33 = k\uj% where [<£*)", i = 0 , .

This completes the proof. 16. L='2iundn£33[d] We now choose an o p e r a t o r power w of UN ( i f i t e x i s t s ) . X*=w\ + 0(\). We c o n s t r u c t a r o o t X of d e g r e e S We d e n o t e by and an AT-admissible r o o t of an JV-th 1° for any S=PN~1GQN = ZAM N of L, as i n Lemma 2 . 1 5 ; t h e element XP£33{(\~1)) . Let i

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Algebraic aspects of nonlinear differential equations by Manin Yu.I.

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