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We give a constructive proof of a global controllability result for anautonomous system of ODEs guided by bounded locally Lipschitz and divergencefree (i.e.\ incompressible) vector field, when the phase space is the wholeEuclidean space and the vector field satisfies so-called vanishing mean driftcondition. For the case when the ODE is defined over some smooth compactconnected Riemannian manifold, we significantly strengthen the assertion of theknown controllability theorem in absence of nonholonomic constraints by provingthat one can find a control steering the state vector from one given point toanother by using the observations of only the state vector, i.e., in otherwords, by changing slightly the vector field, and such a change can be madesmall not only in uniform, but also in Lipschitz (i.e. $C^1$) topology.

Constructive controllability for incompressible vector fields