I have come across the Frobenius theorem in my study of GR, which for the special case of $S^2$ roughly means, that every point of a manifold with spherical symmetry can be foliated by spheres. I know the trivial example of $\mathbb{R}^3$ foliated by spheres.
Can you give me some other example of some other manifold which illustrates this theorem?
Actually, $R^3$ cannot be foliated by spheres; it is $R^3$ minus a point that admits such a foliation (so your theorem is wrong). For other foliated examples:
Take $S^2\times S^1$.
More interestingly, let $f: S^2\to S^2$ be the antipodal map. Now, take $S^2\times [0,1]$ and identify the spheres $S^2\times 0, S^2\times 1$ via the map $F(x,0)=(f(x),1)$. The result is again foliated by 2-spheres. This examples is called a nontrivial $S^2$-bundle over the circle.
The Frobenius theorem, of course, is correct, but it means something different. This theorem gives necessary and sufficient conditions for the given $k$-plane distribution on a smooth manifold $M$ to be the tangent distribution of a smooth foliation on $M$. This theorem cannot predict the topology of the leaves of the foliation.