On the riemannian metrics in which admit all hyperplanes as minimal hypersurfaces

Inspired by a result of Bekkar (1991), Robert Lutz raised the following problem: determine the riemannian metrics in domains of ⁿ which admit all hyperplanes as minimal hypersurfaces. We solve the problem giving a formula which expresses its solutions in terms of the non-degenerate quadratic first integrals of the geodesic motion in the euclidean space (second-order Killing tensor fields). Then, we prove that for n = 3 the non-flat polynomial solutions of the problem are the left invariant riemannian metrics on the Heisenberg group.