  In this note, we show that a nontrivial, compact, degenerate or
nondegenerate, gradient Einstein-type manifold of constant scalar curvature is
isometric to the standard sphere with a well defined potential function.
Moreover, under some geometric assumptions, the noncompact case is also
treated. In this case, the main result is that a homogeneous, proper,
noncompact, nondegenerate, gradient Einstein-type manifold is an Einstein
manifold.
