Is it possible to have  a 2  dimensional foliation of $\mathbb{R}^{3}-\{0\}$ such that each leaf is homeomorphic to the torus? what algebraic topological obstruction exist?

Another question: is there a foliation as above with the following additional property :
The foliation is stable at the origin. that is for every neighborhood V of origin there is a smaller neighborhood W such that the  saturation of W is contained in V? 

the motivation for this question is the concept of "Blow up" of  singularities of vector field. when we blow up a singularity in R^3, we replace the singularity by  a S^2. The  blow up processes is  based on  the fact that R^3-{0} is  foliated by a familly of S^2.  Now if the answer of the above question is positive we can obtain a new type of torus- blow up.
However the blowing up was my main motivation for this question, but ,by this question, I never mean "is R^3-{0} homeomorphic to R x tori?"