You have not been very specific about boundary conditions. Still, the constant mean curvature surfaces (they are not called minimal if the mean curvature is nonzero) of revolution in $\mathbb R^3$ are the circular cylinder and the Delaunay surfaces,
http://en.wikipedia.org/wiki/Constant-mean-curvature_surface 

If your boundary conditions are what i think, called "free boundary," the requirement would be that the surface meet both planes orthogonally.