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On the linearization of vector fields on a torus with prescribed frequency

In this paper we are mainly concerned with the linearization of the flow with prescribed frequency for analytic perturbation of constant vector fields on a torus under weaker non-degeneracy condition and non-resonant condition. As is well known the perturbation of constant vector fields may induce a shift of frequency, when Kolmogorov's non-degeneracy condition is violated. By introducing external parameters and using the polynomial structure to truncate, we prove that if the frequency mapping has the nonzero Brouwer's topological degree at some non-resonant frequency, then the conjugated vector fields will have a linear flow with this frequency.