Infinitely Dimensional Ordinary Differential Equations of a General Kind

We develop a full theory of two classes of infinitely dimensional Hamiltonian systems, containing equations with the potential of Frenkel-Kontorova, Fermi-Pasta-Ulam and their perturbations of a general kind. It includes finding families of periodic solutions and of traveling solutions, creating the infinite-dimensional analog of the Kolmogorov-Moser-Arnol'd theory (the KAM) theory for families of quasiperiodic solutions with continual and countable and finite number of rational independent frequencies. We also prove stability of stationary solutions. The last statement contradicts to the Arnol'd hypothesis on unstability in Hamiltonian systems. The paper is a continuation of preprints №50 and 55.