  The Hamiltonian structure of spacetimes with two commuting Killing vector
fields is analyzed for the purpose of addressing the various problems of time
that arise in canonical gravity. Two specific models are considered: (i)
cylindrically symmetric spacetimes, and (ii) toroidally symmetric spacetimes,
which respectively involve open and closed universe boundary conditions. For
each model canonical variables which can be used to identify points of space
and instants of time, {\it i.e.}, internally defined spacetime coordinates, are
identified. To do this it is necessary to extend the usual ADM phase space by a
finite number of degrees of freedom. Canonical transformations are exhibited
that identify each of these models with harmonic maps in the parametrized field
theory formalism. The identifications made between the gravitational models and
harmonic map field theories are completely gauge invariant, that is, no
coordinate conditions are needed. The degree to which the problems of time are
resolved in these models is discussed.
