Analytic conjugation between planar differential systems and potential systems

In this article it is proved that an analytical planar vector field with a non-degenerate center at \((0,0)\) is analytically conjugate, in a neighborhood of \((0,0)\), to a Hamiltonian vector field of the form \(y\frac{\partial}{\partial x}-V'(x)\frac{\partial}{\partial y}\), where \(V\) is an analytic function defined in a neighborhood of the origin such that \(V(0)=V'(0)=0\) and \(V''(0)>0.\)