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

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.