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In this paper, we introduce some new results on the study of Z q -equivariant planar polynomial vector fields. The main conclusions are as follows.(1) For the Z 2 -equivariant planar cubic systems having two elementary foci, the characterization of a bi-center problem and shortened expressions of the first six Lyapunov constants are completely solved. The necessary and sufficient conditions for the existence of the bi-center are obtained. All possible first integrals are given. Under small Z 2 -equivariant cubic perturbations, the conclusion that there exist at most 12 small-amplitude limit cycles with the scheme 6 6 is proved. (2) On the basis of mentioned work in (1), by considering the bifurcation of a global limit cycle from infinity, we show that under small Z 2 -equivariant cubic perturbations, such bi-center cubic system has at least 13 limit cycles with the scheme 1 6 6 , i.e., we obtain that the Hilbert number H (3) ≥ 13. (3) For the Z 5 -equivariant planar polynomial vector field of degree 5, we shown that such system has at least five symmetric centers if and only if it is a Hamltonian vector field. The characterization of a center problem is completely solved. The shortened expressions of the first four Lyapunov constants are given. Under small Z 5 -equivariant perturbations, the conclusion that perturbed system has

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