The Hénon-Heiles Hamiltonian with a cubic potential and an additive non polynomial term is known to be Liouville integrable in 3 cases, respectively associated with the stationary flow of the fifth-KdV, Sawada-Kotera(SK) and Kaup-Kupershmidt partial differential equations. Here, we integrate in the 3 cases the equations of motion with genus 2 hyperelliptic functions. This extends previous results obtained for the SK and KK cases. [<6>]