  We consider the Zakharov system in the energy critical dimension $d=4$ with
energy below the ground state. It is known that below the ground state
solutions exist globally in time, and scatter in the radial case. Scattering
below the ground state in the non-radial case is an open question. We show that
if scattering fails, then there exists a minimal energy non-scattering solution
below the ground state. Moreover the orbit of this solution is precompact
modulo translations. The proof follows by a concentration compactness argument,
together with a refined small data theory for energy dispersed solutions.
