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In this paper we consider the persistence of lower dimensional tori of a class of analytic perturbed Hamiltonian system, H =hw(x), Ii + 1 W0 (u 2 + v 2 ) + P(q, I, z, ¯ z;x) and prove that if the frequencies (w0,W0) satisfy some non-resonance condition and the Brouwer degree of the frequency mapping w(x) at w0 is nonzero, then there exists an invariant lower dimensional invariant torus, whose frequencies are a small dilation of w0.

The persistence of elliptic lower dimensional tori with prescribed frequency for Hamiltonian systems