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We show that a dynamically convex Reeb flow on a lens space $L(p, 1), p>1,$ admits a $p$-unknotted closed Reeb orbit $P$ which is the binding of a rational open book decomposition with disk-like pages. Each page is a rational global surface of section for the Reeb flow and the Conley-Zehnder index of the $p$-th iterate of $P$ is $3$. This result applies to the H\'enon-Heiles Hamiltonian whose energy level presents $\mathbb{Z}_3$-symmetric and for all energies $<1/6$ the flow restricted to the sphere-like component descends to a dynamically convex Reeb flow on $L(3,1)$. Due to a $\mathbb{Z}_4$-symmetry the result also applies to Hill's lunar problem.

Global surfaces of section for dynamically convex Reeb flows on lens spaces