  In this paper we study the problem of prescribing the $\bar
Q^{\prime}$-curvature on pseudo-Einstein CR 3-manifolds. In the first stage we
study the problem in the compact setting and we show that under natural
assumptions, one can prescribe any positive CR pluriharmonic function. In the
second stage we study the problem in the non-compact setting of the Heisenberg
group. Under mild assumptions on the prescribed function, we prove the
existence of a one parameter family of solutions. In fact, we show that one can
find two kinds of solutions: normal ones that satisfy an isoperimetric
inequality and non-normal ones that have a biharmonic leading term.
