  Let $X$ be a polynomial vector field in $\mathbb{R}^2$ which, after one-point
compactification of the plane, has a punctured neighbourhood $\dot U$ of the
point at infinity which is foliated by closed orbits of $X$. If the period
function of $X$ in $\dot U$ is bounded from below by a positive constant, $X$
is necessarily linear, hence conjugated, up to a nonzero constant factor, to
$-y \frac{\partial}{\partial x} + x \frac{\partial}{\partial y}$. This result
answers to a question posed by J. Llibre , proving e.g. that a global
isochronous center is linear.
