Rotational Ricci surfaces

We classify rotational surfaces in the three-dimensional Euclidean space whose Gaussian curvature K satisfies \(\begin{aligned} K\Delta K - \Vert \nabla K\Vert ^2-4K^3 = 0. \end{aligned}\) $$\begin{array}{c}\hfill K\mathrm{\Delta }K-{\Vert \mathrm{\nabla }K\Vert }^2-4K^3=0.\end{array}$$ 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.