A KAM THEOREM OF DEGENERATE INFINITE-DIMENSIONAL HAMILTONIAN-SYSTEMS .1.

A class of weaker nondegeneracy conditions is given and an existence t heorem of invariant tori is proven for small perturbations of degenera te integrable infinite dimensional Hamiltonian systems under the weake r nondegeneracy conditions. The measure estimates of the parameter set are also given for which invariant tori exist. It is valuable to poin t out that by the motivation of finite dimensional situation the nonde generacy conditions may be the weakest. Mainly KAM machine is used to prove the existence of invariant tori. The measure estimates for small divisor conditions, on which the measure estimates of the parameter s et are based, will be given in the second paper.