A renormalization approach has been used in \cite{EKW1} and \cite{EKW2} to
prove the existence of a \textit{universal area-preserving map}, a map with
hyperbolic orbits of all binary periods. The existence of a horseshoe, with
positive Hausdorff dimension, in its domain was demonstrated in \cite{GJ1}. In
this paper the coexistence problem is studied, and a computer-aided proof is
given that no elliptic islands with period less than $18$ exist in the domain.
It is also shown that the area enclosed by elliptic islands is less than
$0.046$. This is highly unexpected, since generically it is believed that for
conservative systems hyperbolicity and ellipticity coexist.