The inverse scattering method for a relativistic string

A relativistic string is described by the inverse scattering method for periodic potentials. The auxiliary spectral problem and the analytic properties of the Bloch functions are investigated. A complete set of involutive relativistieally invariant integrals of the motion is obtained. Equations are obtained for string configurations that have a finite number of gaps in the spectrum of the auxiliary problem. It is shown that such a condition is equivalent to stationarity of the configurations with respect to the higher string equations. The quantum theory in the single-gap sector is discussed.