  Foliate systems are those which preserve some (possibly singular) foliation
of phase space, such as systems with integrals, systems with continuous
symmetries, and skew product systems. We study numerical integrators which also
preserve the foliation. The case in which the foliation is given by the orbits
of an action of a Lie group has a particularly nice structure, which we study
in detail, giving conditions under which all foliate vector fields can be
written as the sum of a vector field tangent to the orbits and a vector field
invariant under the group action. This allows the application of many
techniques of geometric integration, including splitting methods and Lie group
integrators.
