  It is shown explicitly that when the characteristic vector field that defines
a Godel-type metric is also a Killing vector, there always exist closed
timelike or null curves in spacetimes described by such a metric. For these
geometries, the geodesic curves are also shown to be characterized by a lower
dimensional Lorentz force equation for a charged point particle in the relevant
Riemannian background. Moreover, two explicit examples are given for which
timelike and null geodesics can never be closed.
