  Negative kinetic energies correspond to ghost degrees of freedom, which are
potentially of relevance for cosmology, quantum gravity, and high energy
physics. We present a novel wide class of stable mechanical systems where a
positive energy degree of freedom interacts with a ghost. These theories have
Hamiltonians unbounded from above and from below, are integrable, and contain
free functions. We show analytically that their classical motion is bounded for
all initial data. Moreover, we derive conditions allowing for Lyapunov stable
equilibrium points. A subclass of these stable systems has simple polynomial
potentials with stable equilibrium points entirely due to interactions with the
ghost. All these findings are fully supported by numerical computations which
we also use to gather evidence for stability in various nonintegrable systems.
