Global solvability for first order real linear partial differential operators II

Let L be a real C∞ vector field on a smooth manifold X, vanishing at exactly one point x0. From the pioneering work of B. Malgrange (1955–1956) [6], we know that solvability of P=L+c on C∞(X), for c∈C∞(X,C), implies that: (a) X is L-convex. Also, it follows: (b) a non-resonance condition for the jet-solvability at x0.In a previous paper, in addition to (a) and (b), the authors showed that P is globally solvable on C∞ if we assume: (c) a non-resonance condition in order to linearize L near x0; that (d) the only relatively compact orbit of L is {x0}; and that (e) c is real.Here we obtain the same conclusion without (c) and (e).