  A Hamilton-Jacobi theory for general dynamical systems, defined on fibered
phase spaces, has been recently developed. In this paper we shall apply such a
theory to contact Hamiltonian systems, as those appearing in thermodynamics and
on geodesic flows in fluid mechanics. We first study the partial and complete
solutions of the Hamilton-Jacobi Equation (HJE) related to these systems. Then
we show that, for a given contact system, the knowledge of what we have called
a complete pseudo-isotropic solution ensures the integrability by quadratures
of its equations of motion. This extends to contact manifolds a recent result
obtained in the context of general symplectic and Poisson manifolds.
