On a theorem by Ekeland-Hofer

In EH89, Theorem 1] Ekeland-Hofer prove that for a centrally symmetric, restricted contact type hypersurface in ℝ²ⁿ and for any global, centrally symmetric Hamiltonian perturbation there exists a leaf-wise intersection point. In this note we show that if we replace restricted contact type by star-shaped, there exist infinitely many leaf-wise intersection points or a leaf-wise intersection point on a closed characteristic.