Global normalizations for centers of planar vector fields

This paper addresses a question posed by Carmen Chicone and proves that an analytic vector field with a non-degenerate global center can be transformed into a classical Newtonian equation $\ddot{x}=-V^{\prime }\left(x\right)$.

Additionally, we establish a global Poincaré normal form for planar centers. We also demonstrate the global analytic integrability of the equation $\ddot{x}=F(u,\dot{u})$, where $F(u,v)=F(u,-v)$, under some additional conditions.