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Is it possible to have a 2 dimensional foliation of 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?"

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?"

Is it possible to have a 2 dimensional foliation of 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.

Is it possible to have a 2 dimensional foliation of 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?"