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The aim of this paper is to give a sufficient and necessary condition of thegeneralized polynomial Li\'enard system with a global center (including lineartyper and nilpotent type). Recently, Llibre and Valls [J. DifferentialEquations, 330 (2022), 66-80] gave a sufficient and necessary condition of thegeneralized polynomial Li\'enard system with a linear type global center. It iseasy to see that our sufficient and necessary condition is more easy bycomparison. In particular, we provide the explicit expressions of all thegeneralized polynomial Li\'enard differential systems of degree 5 having aglobal center at the origin and the explicit expression of a generalizedpolynomial Li\'enard differential system of indefinite degree having a globalcenter at the origin.

A sufficient and necessary condition of generalized polynomial Li\'enard systems with global centers