On one conjecture in the theory of isochronous Liénard systems (Q1701765)

Theorem: The real change of variables \[ u=x+ \sum^\infty_{k=2} \alpha_k x^k,\quad v=y+ \sum^\infty_{k=2} \beta_k x^k\tag{1} \] takes the holomorphic Liénard system \[ \dot x=-y,\;\dot y=g(x)- f(x)y,\;g(0)= 0,\;g'(0)=-1,\;f(0)=0\tag{2} \] to a system of the form \[ \dot u=-\Biggl(v+ u+ \sum^\infty_{k=2} \gamma_{k-1} u^{k-1}\Biggr)\,d(u),\quad \dot v=\Biggl(u-v \sum^\infty_{k=2} \gamma_{k-1} u^{k-1}\Biggr)\, d(u),\tag{3} \] where \(d(u)= \left(1+ \sum^\infty_{k=2} H_k u^{k-1}\right)^{-1}\). A necessary and sufficient condition for the singular point \(0(0,0)\) of system (2) to be a center is that there exists a real transformation (1) taking system (2) to a system (3), where \(\gamma_{2k}= 0\), \(k=1,2,\dots\). A necessary and sufficient condition for the singular point \(0(0,0)\) of system (2) to be an isochronous center is that there exist a real transformation (1) taking system (2) to a system (3), where \(\gamma_{2k}= 0\), \(k=1,2,\dots, H_{2k-1}= 0\), \(k=1, 2,\dots\).