  We consider an integrable scalar partial differential equation (PDE) that is
second order in time. By rewriting it as a system and applying the
Wahlquist-Estabrook prolongation algebra method, we obtain the zero curvature
representation of the equation, which leads to a Lax representation in terms of
an energy-dependent Schr\"{o}dinger spectral problem of the type studied by
Antonowicz and Fordy. The solutions of this PDE system, and of its associated
hierarchy of commuting flows, display weak Painlev\'e behaviour, i.e. they have
algebraic branching. By considering the travelling wave solutions of the next
flow in the hierarchy, we find an integrable perturbation of the case (ii)
Henon-Heiles system which has the weak Painlev\'{e} property. We perform
separation of variables for this generalized Henon-Heiles system, and describe
the corresponding solutions of the PDE.
