  In this paper we study the transformations linearizing isochronous centers of
planar Hamiltonian differential systems with polynomial Hamiltonian functions
$H(x,y)$ having only isolated singularities. Assuming the origin is an
isochronous center lying on the level curve $L_0$ defined by $H(x,y)=0$, we
prove that, there exists a canonical linearizing transformation analytic on a
simply-connected open set $\Omega$ with closure
$\overline{\Omega}=\mathbb{R}^2$, if and only if,
  $L_0$ consists of only isolated points; furthermore, if the origin is the
unique center, then the condition that $L_0$ consists of only isolated points
implies that the corresponding canonical linearizing transformation can be
analytically defined on the whole plane.
