  The three integrable two-dimensional Henon-Heiles systems and their
integrable perturbations are revisited. A family of new integrable
perturbations is found, and N-dimensional completely integrable generalizations
of all these systems are constructed by making use of sl(2,R)+h(3) as their
underlying Poisson symmetry algebra. In general, the procedure here introduced
can be applied in order to obtain N-dimensional integrable generalizations of
any 2D integrable potential of the form V(q_1^2, q_2), and the formalism gives
the explicit form of all the integrals of the motion. Further applications of
this algebraic approach in different contexts are suggested.
