  In this paper, we investigate the existence of KAM tori for an infinite
dimensional Hamiltonian system with finite number of zero normal frequencies.
By constructing a constant quantity we show that, for "most" frequencies in the
sense of Lebesgue measure, either if the quantity is zero, there is a KAM tori
or if the quantity is not zero, there is no KAM tori in some domain. As
application, we show that the nonlinear Schr\"{o}dinger equation with a zero
frequency possesses many quasi-periodic solutions.
