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 $y\frac{\mathrm{\partial }}{\mathrm{\partial }x}-V^{\mathrm{\prime }}(x)\frac{\mathrm{\partial }}{\mathrm{\partial }y}$, where $V$ is an analytic function defined in a neighborhood of the origin such that $V(0)=V^{\mathrm{\prime }}(0)=0$ and $V^{\mathrm{\prime }\mathrm{\prime }}(0)>0.$