On Chouikha's isochronicity criterion

Recently A.R.Chouikha gave a new characterization of isochronicity of center at the origin for the equation $x\mathrm{"}+g(x)=0$, where $g$ is a real smooth function defined in some neighborhood of $0\in \mathbb{R}$. We describe another proof of his characterization and some new development of the subject.