Inspired by this question on homothety vector fields we realize that non homotheticity is some how an intrinsic property of the foliation associated to the vector field. See the comment by Prof. Bryant $x^2\partial_x+xy\partial_y$. However it is a singular foliation but the dynamical feature is an obstruction for homothecity since the origin attracts the left part $x<0$ and it repels the right part $x>0$. The non singular case is mentioned in my answer to that question indicating to semi stable limit cycle, very simillar concept to the interesting situation in the comment of Prof. Bryant.
So the dynamical obstruction mentioned above is a motivation to ask the following generalized question:
Assume that we have a 2 dimensional foliation of $S^3$. Are there two independent homothety vector fields tangent to the foliation(Homothety with respect to some volume form)? A weaker version: Is there a homothety vector field (wrt some volum form) tangent to the foliation?
