The Hopf fibration $S^1\rightarrow S^3\rightarrow S^2$ gives a decomposition of $S^3$ into 2-tori and to circles, so that the tori are foliated by circles of slope 1. If you take the region between two of this tori and identified by an homeomorphism that preserves the foliation you get a foliation by circles on the 3 torus, which I believe is equivalent to the foliation of $S^1$ on $S^1\times S^1\times S^1$ given by its product structure. Is this true?
Also if you remove the two circles of the decomposition I mentioned you get a fibration $S^1\rightarrow S^1\times S^1\times \mathbb R\rightarrow \ S^1\times \mathbb R$ which I believe is given by the multiplication in the first entry and projection in the other two. Is this also true?