  Static, axisymmetric solutions form a large class of important black holes in
classical GR. In four dimensions, the existence of their most general metric
ansatz relies on the fact that two-dimensional subspaces of the tangent space
at each point spanned by vectors orthogonal to the time-translation and
rotation Killing fields are integrable. This was first proved by Wald via an
application of Frobenius theorem. In this note, we furnish an elementary proof
for this theorem by Wald in arbitrary dimensions which yields the metric ansatz
for the most general solution of the D-dimensional vacuum Einstein equations
that admits D-2 orthogonal and commuting Killing vector fields.
