  In this paper we consider axisymmetric black holes in supergravity and
address the general issue of defining a first order description for them. The
natural setting where to formulate the problem is the De
Donder-Weyl-Hamilton-Jacobi theory associated with the effective
two-dimensional sigma-model action describing the axisymmetric solutions. We
write the general form of the two functions S_m defining the first-order
equations for the fields. It is invariant under the global symmetry group G_(3)
of the sigma-model. We also discuss the general properties of the solutions
with respect to these global symmetries, showing that they can be encoded in
two constant matrices belonging to the Lie algebra of G_(3), one being the
Noether matrix of the sigma model, while the other is non-zero only for
rotating solutions. These two matrices allow a G_(3)-invariant characterization
of the rotational properties of the solution and of the extremality condition.
We also comment on extremal, under-rotating solutions from this point of view.
