Strong isochronicity of the Liénard system (Q2371634)

Consider the Liénard system \[ \dot x=- y,\quad \dot y= a(x)- b(x) y,\tag{\(*\)} \] where \(a\) and \(b\) are holomorphic functions given by \[ a(x)= x+\sum^\infty_{i=2} a_i x^i,\quad b(x)= \sum^\infty_{j=1} b_j x^j. \] The author derives conditions on the coefficients \(a_i\) and \(b_i\) which are equivalent for the equilibrium \(x= y= 0\) to be an isochronous center of \((*)\).