In this note we prove the following result:
Let X be a compact complex manifold, on which the group G=(C,+) acts holomorphically. Assume that there exists a G-invariant subset U of positive Lebesgue measure in which every G-orbit is compact one-dimensional.
Then the G-action fibers through a torus action, ie G=(C,+) contains a lattice H which acts trivially on X. In particular all orbits in X are compact one-dimensional tori isogeneous to each other.
manuscripta math. 82, 89--91 (1994)