Integration of a generalized Hénon-Heiles Hamiltonian

The generalized Henon–Heiles Hamiltonian H=1/2(PX2+PY2+c1X2+c2Y2)+aXY2−bX3/3 with an additional nonpolynomial term μY−2 is known to be Liouville integrable for three sets of values of (b/a,c1,c2). It has been previously integrated by genus two theta functions only in one of these cases. Defining the separating variables of the Hamilton–Jacobi equations, we succeed here, in the two other cases, to integrate the equations of motion with hyperelliptic functions.