Stability of compact actions of the Heisenberg group

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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 M⁴ is given by linearly independent vector fields X₁, X₂, X₃ such that X₃ = [X₁,X₂] and [X₁,X₃] = [X₂, X₃] = 0. The C¹ topology for vector fields induces a topology in the space of actions of G on M⁴. An action is compact if all orbits are compact. Given a compact action θ, we investigate under which conditions its C¹ perturbations θ 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 Rⁿ, the definitions do not turn out to be equivalent.