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The existence of lower dimensional KAM tori is shown for a class of nearlyintegrable Hamiltonian systems where the second Melnikov's conditions areeliminated. As a consequence, it is proved that there exist many invariant toriand thus quasi-periodic solutions for nonlinear wave equations, Schr\"odingerequations and other equations of higher spatial dimension

A KAM Theorem with Applications to Partial Differential Equations of Higher Dimensions