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The fermionic and bosonic zero modes of the one-dimensional (1D) interacting Kitaev chain at the symmetric point are unveiled.The many-body structures of the Majorana zero modes in the topological region are given explicitly by carrying out a perturbation expansion up to infinite order.We also give the analytic expressions of the bosonic zero modes in the topologically trivial phase.Our results are generalized to the hybrid fermion system comprised of the interacting Kitaev model and the Su-Schrieffer-Heeger (SSH) model,in which we show that these two types of zero modes can coexist in a certain region of its phase diagram.