  We classify rotational surfaces in the three-dimensional Euclidean space
whose Gaussian curvature $K$ satisfies
  \begin{equation*}
  K\Delta K - \|\nabla K\|^2-4K^3 = 0.
  \end{equation*}
  These surfaces are referred to as rotational Ricci surfaces. As an
application, we show that there is a one-parameter family of such surfaces
meeting the boundary of the unit Euclidean three-ball orthogonally. In
addition, we show that this family interpolates a vertical geodesic and the
critical catenoid.
