  In this paper, we investigate conformal Killing's vectors (CKVs) admitted by
some plane symmetric spacetimes. Ten conformal Killing's equations and their
general forms of CKVs are derived along with their conformal factor. The
existence of conformal Killing's symmetry imposes restrictions on the metric
functions. The conditions imposing restrictions on these metric functions are
obtained as a set of integrability conditions. Considering the cases of
time-like and inheriting CKVs, we obtain spacetimes admitting plane conformal
symmetry. Integrability conditions are solved completely for some known
non-conformally flat and conformally flat classes of plane symmetric
spacetimes. A special vacuum plane symmetric spacetime is obtained, and it is
shown that for such a metric CKVs are just the homothetic vectors (HVs). Among
all the examples considered, there exists only one case with a six dimensional
algebra of special CKVs admitting one proper CKV. In all other examples of
non-conformally flat metrics, no proper CKV is found and CKVs are either HVs or
Killing's vectors (KVs). In each of the three cases of conformally flat
metrics, a fifteen dimensional algebra of CKVs is obtained of which eight are
proper CKVs.
