Prove of homotopy equivalence using differential equation

Question

I have to prove that $R^3 \setminus L_1,...,L_n $, where $L_1,..,L_n$ are non intersecting lines, is homotopy equivalent to a wedge sum of $n$ circles. So once I've managed to show that it is possible to move them [the lines] homeomorphically in such a manner that they all lie on one plain it was easy to reach conclusion. But my argument concerning the existence of homeomorphism was rather awful.Anyway, I've heard it is possible to do it nicely using differential equations (maybe something connected with Hartman-Grobman theorem?). I'll be grateful for any hints how to do it that way.