Geodesic
The group of internal symmetries and the conditions of integrability of two-dimensional dynamical systems
Teoretičeskaâ i matematičeskaâ fizika, Tome 51 (1982) no. 1, pp. 10-21 
A. N. Leznov; V. G. Smirnov; A. B. Shabat. The group of internal symmetries and the conditions of integrability of two-dimensional dynamical systems. Teoretičeskaâ i matematičeskaâ fizika, Tome 51 (1982) no. 1, pp. 10-21. http://geodesic.mathdoc.fr/item/TMF_1982_51_1_a1/
@article{TMF_1982_51_1_a1,
     author = {A. N. Leznov and V. G. Smirnov and A. B. Shabat},
     title = {The~group of internal symmetries and the conditions of integrability of two-dimensional dynamical systems},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {10--21},
     year = {1982},
     volume = {51},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1982_51_1_a1/}
}
TY  - JOUR
AU  - A. N. Leznov
AU  - V. G. Smirnov
AU  - A. B. Shabat
TI  - The group of internal symmetries and the conditions of integrability of two-dimensional dynamical systems
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 1982
SP  - 10
EP  - 21
VL  - 51
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TMF_1982_51_1_a1/
LA  - ru
ID  - TMF_1982_51_1_a1
ER  - 
%0 Journal Article
%A A. N. Leznov
%A V. G. Smirnov
%A A. B. Shabat
%T The group of internal symmetries and the conditions of integrability of two-dimensional dynamical systems
%J Teoretičeskaâ i matematičeskaâ fizika
%D 1982
%P 10-21
%V 51
%N 1
%U http://geodesic.mathdoc.fr/item/TMF_1982_51_1_a1/
%G ru
%F TMF_1982_51_1_a1

Voir la notice de l'article provenant de la source Math-Net.Ru

The concept of the characteristic algebra of a system of equations of the form $u_{z\overline{z}}=F(u)$ is introduced. This algebra is associated with Lie–Bäcklund transformations. The conditions of integrability of such systems are formulated. It is shown that the case of integrability in quadrature corresponds to finite dimensionality of the characteristic algebra, while the case of integrability by the inverse scattering technique corresponds to this algebra's having a finite-dimensional representation. These requirements determine the form of the right-hand side $F$ for integrable systems.

[1] Zhiber A. V., Shabat A. B., DAN SSSR, 247:5 (1979), 1103–1105 | MR

[2] Zhiber A. V., Ibragimov N. Kh., Shabat A. B., DAN SSSR, 249:1 (1979), 26–29 | MR | Zbl

[3] Leznov A. N., TMF, 42:3 (1980), 343–349 | MR | Zbl

[4] Leznov A. N., Saveliev M. V., Lett. Math. Phys., 3 (1973), 489–494 ; Commun. Math. Phys., 74 (1980), 111–118 | DOI | MR | DOI | MR | Zbl

[5] Kats V. G., Izv. AN SSSR, ser. matem., 32 (1968), 1323–1367 | MR | Zbl

[6] Ibragimov N. Kh., Shabat A. B., Funkts. analiz i ego prilozh., 14:4 (1980), 79–80 | MR | Zbl

[7] Sokolov V. V., Shabat A. B., Funkts. analiz i ego prilozh., 14:2 (1980), 79–81 | MR

[8] Zakharov V. E., Shabat A. B., Funkts. analiz i ego prilozh., 13:3 (1979), 13–23 ; Михайлов А. В., Письма ЖЭТФ, 30 (1979), 443–447; Захаров В. Е., Манаков С. В., Новиков С. П., Питаевский Л. П., Теория солитонов: Метод обратной задачи рассеяния, Наука, М., 1980 | MR | MR | Zbl

[9] Inönü E., Wigner E. P., Proc. Acad. Sci. USA, 39 (1956), 510–518 | DOI | MR

[10] Dzhekobson N., Algebry Li, Mir, M., 1964 | MR