This article is cited in 73 papers

The group of internal symmetries and the conditions of integrability of two-dimensional dynamical systems

A. N. Leznov, V. G. Smirnov, A. B. Shabat


Abstract: 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.

Received: 23.01.1981

 English version: Theoretical and Mathematical Physics, 1982, 51:1, 322–330

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