Poincare: Change of form to a primitive 1-form

Question

Given a 2-form w on $R^3$, such that dw=0; how does one find all 1-forms k such that dk=w?

Provided a homotopy H(t) one can pull back w and apply the Poincare operator on it. But what if one does not have a homotopy to start working with?

Answer 1

If your form is defined on the whole $\mathbb{R}^3$, then there is always the linear homotopy which contracts every point to the origin.