The index of stable critical points

In this paper we show that in dimension greater or equal than 3 the index of a stable critical point can be any integer. More concretely, given any k∈Z and n⩾3 we construct a C∞ vector field on Rn with a unique critical point which is stable (in positive and negative time) and has index equal to k. This result extends previous ones on the index of stable critical points.