  Following the basic principles stated by Painlev\'e, we first revisit the
process of selecting the admissible time-independent Hamiltonians
$H=(p_1^2+p_2^2)/2+V(q_1,q_2)$ whose some integer power $q_j^{n_j}(t)$ of the
general solution is a singlevalued function of the complex time $t$. In
addition to the well known rational potentials $V$ of H\'enon-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 H\'enon-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.
