Differential-geometric structure and spectral properties of nonlinear completely integrable dynamical systems of the Mel'nikov type

One considers V. K. Mel'nikov's new class of nonlinear dynamical systems, which is a generalization of the Korteweg-de Vries dynamical system. One investigates the differential-geometric and spectral properties of dynamical systems of Mel'nikov type, one gives their Hamiltonian form, one establishes the so-called gradient identity. The class of finite-zone potentials of a Sturm-Liouville operator, satisfying the given dynamical systems, is described.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 5, pp. 655–659, May, 1990.