  In this article, we study the possibility of sustaining a static and
spherically symmetric traversable wormhole geometries admitting conformal
motion in Einstein gravity, which presents a more systematic approach to search
a relation between matter and geometry. In wormhole physics, the presence of
exotic matter is a fundamental ingredient and we show that this exotic source
can be dark energy type which support the existence of wormhole spacetimes. In
this work we model a wormhole supported by dark energy which admits conformal
motion. We also discuss the possibility of detection of wormholes in the outer
regions of galactic halos by means of gravitational lensing. The studies of the
total gravitational energy for the exotic matter inside a static wormhole
configuration are also done.
