  Exact solutions of traversable wormholes are found under the assumption of
spherical symmetry and the existence of a {\it non-static} conformal symmetry,
which presents a more systematic approach in searching for exact wormhole
solutions. In this work, a wide variety of solutions are deduced by considering
choices for the form function, a specific linear equation of state relating the
energy density and the pressure anisotropy, and various phantom wormhole
geometries are explored. A large class of solutions impose that the spatial
distribution of the exotic matter is restricted to the throat neighborhood,
with a cut-off of the stress-energy tensor at a finite junction interface,
although asymptotically flat exact solutions are also found. Using the ``volume
integral quantifier,'' it is found that the conformally symmetric phantom
wormhole geometries may, in principle, be constructed by infinitesimally small
amounts of averaged null energy condition violating matter. Considering the
tidal acceleration traversability conditions for the phantom wormhole geometry,
specific wormhole dimensions and the traversal velocity are also deduced.
