  We use the Fuchsian algorithm to study the behavior near the singularity of
certain families of U(1) Symmetric solutions of the vacuum Einstein equations
(with the U(1) isometry group acting spatially). We consider an analytic family
of polarized solutions with the maximum number of arbitrary functions
consistent with the polarization condition (one of the ``gravitational degrees
of freedom'' is turned off) and show that all members of this family are
asymptotically velocity term dominated (AVTD) as one approaches the
singularity. We show that the same AVTD behavior holds for a family of ``half
polarized'' solutions, which is defined by adding one extra arbitrary function
to those characterizing the polarized solutions. (The full set of non-polarized
solutions involves two extra arbitrary functions). We begin to address the
issue of whether AVTD behavior is independent of the choice of time foliation
by showing that indeed AVTD behavior is seen for a wide class of choices of
harmonic time in the polarized and half-polarized (U(1) Symmetric vacuum)
solutions discussed here. \
