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Extra info for Communications In Mathematical Physics - Volume 298
2. It was done by use of what we call the Kotter trick (see [20,24]). The nature of this transformation is going to be clarified in Sect. 5 through the theory of multivalued groups. Then, we manage to generalize another Kotter transformation and this gives us a possibility to integrate the general system defined at the beginning of this section. We reduce the problem to the functions Pi , i = 1, 2, 3. The evolution of those functions in terms of the theta-functions was obtained by Kowalevski herself in .
We want to find points N1 = (y1 , v1 ) and N2 = (y2 , v2 ) on to P and M. These points are 2 which correspond by F2 −V (s, x) + 4nu 2x T (s, y1 ) + V (s, y1 ) , v1 = − , 2T (s, x) 4n −V (s, x) − 4nu 2x T (s, y2 ) + V (s, y2 ) , v2 = − . y2 = 2T (s, x) 4n y1 = By trivial algebraic transformations −4mx 2 − 4xm 2 + xg2 + mg2 + 2g3 + 2nu −4(x − m)2 −4mx(x + m) + x 3 + m 3 − x 3 + xg2 + g3 − m 3 + mg2 + g3 + 2nu = −4(x − m)2 y1 = = −x − m + u−n 2(x − m) 2 , we get the first part of the operation of the two-valued group ( 2 , Z2 ) defined by the relation (47).
If the coefficient K is nonzero we may normalize it to be equal to one. Under this assumption, Eqs. (23) with K = 1 are general. The case K = 0 is going to be analyzed separately in one of the following sections. From Eqs. (22) we get the following Corollary 4. The relation e2 P(x1 ) + e1 P(x2 ) − H (x1 , x2 ) + k 2 (x1 − x2 )2 = 0, (26) is satisfied, where P is the polynomial defined in Lemma 4. Corollary 5. The differentials of x1 and x2 may be written in the form d x1 = −β P(x1 ) + e1 (x1 − x2 )2 , dt d x2 = β P(x2 ) + e2 (x1 − x2 )2 .