Hamiltonians with two degrees of freedom admitting a singlevalued general solution

Following the basic principles stated by Painlevé, we first revisit the process of selecting the admissible time-independent Hamiltonians $$H=(p_1^2+p_2^2)/2+V(q_1,q_2),$$ for which the integer power $q_j^{n_j}(t)$ of the general solution is a single-valued function of the complex time $t$. In addition to the well-known rational potentials $V$ of Hénon-Heiles, this selects possible cases with a trigonometric dependence of $V$ on $q_j$. Then, by establishing the relevant confluences, we restrict the question of the explicit integration of the seven (three `cubic' plus four `quartic') rational Hénon-Heiles cases to the quartic cases. Finally, we perform the explicit integration of the quartic cases, thus proving that the seven rational cases have a meromorphic general solution explicitly given by a genus two hyperelliptic function.