Symmetries of scaler fields. III. Two-dimensional integrable models

The symmetry of the relativistically invariant two-dimensional system $\varphi_{12}=f(\varphi)$ is investigated. Necessary and sufficient conditions for the system to belong to the Liouville type are obtained. Simple examples of Liouville systems are given. It is shown that to establish all integrable systems it is sufficient to assume that the elements of the Lie-Bäicklund algebra are polynomials in $\varphi_i$. Determining equations for the recursion operator of the system are obtained, and two examples of recursion operators are given. Some general arguments about methods of calculating conserved densities are presented.