Lynge (see (4)), proved that X vector field on a compact and connected manifold M, is Killing in a Riemannian metric g, if and only if there exist k complete vector fields X1 , ... , Xk; such that:
1) The fl.ows of the Xi are a11 periodic,
2) [X;, X;]= O, i,j = 1, ... , k,
3) = E,! 1 c; X;, c; real numbers...
