  Let G be the Heisenberg group of real lower triangular 3x3 matrices with unit
diagonal. A locally free smooth action of G on a manifold M^4 is given by
linearly independent vector fields X_1, X_2, X_3 such that X_3 = [X_1,X_2] and
[X_1,X_3] = [X_2, X_3] = 0. The C^1 topology for vector fields induces a
topology in the space of actions of G on M^4. An action is compact if all
orbits are compact. Given a compact action $\theta$, we investigate under which
conditions its C^1 perturbations $\tilde\theta$ are guaranteed to be compact.
There is more than one interesting definition of stability, and we show that in
the case of the Heisenberg group, unlike for actions of R^n, the definitions do
not turn out to be equivalent.
