In this paper we consider the persistence of lower dimensional tori of a class of analytic perturbed Hamiltonian system, and prove that if the frequencies (ω 0 , Ω 0 ) satisfy some non-resonance condition and the Brouwer degree of the frequency mapping ω(ξ) at ω 0 is nonzero, then there exists an invariant lower dimensional invariant torus, whose frequencies are a small dilation of ω 0 . The Hamiltonian equations of motion of N arė Thus for each ξ ∈ D, there exists an invariant n-dimensional

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... us T n × {0} × {0} ⊂ R 2n × R 2 with tangential frequencies ω(ξ), which has an elliptic fixed point in the normal uv-space with normal frequency Ω 0 . These tori are called lower dimensional invariant tori, split from resonant ones lying in the resonance zone constituted by both stochastic trajectories and regular orbits. The persistence of lower dimensional invariant tori has been widely studied. See many significant works [3, 4, 9, 11, 12, 14, 22] . The classical KAM theorem [1, 10, 13] asserts that, under Kolmogorov non-degeneracy condition, namely, det(∂ω/∂p) = 0, if the perturbation is sufficiently small, a Cantor family of n-dimensional Lagrangian invariant tori (so-called maximal dimensional invariant tori) persists with the frequencies ω satisfying Diophantine conditions: