There's a distinction to be made between two notions: *foliations* and *distributions*.

A **distribution** is the data, at each point *m* of *M*, of a subspace of *T*_{m}(*M*). These subspaces are all of the same dimension (say *r*), and depend smoothly on the point *m*, which means that they are generated by *r* smooth vector fields.

A **foliation** is a partition of the manifold into (not necessarily closed) submanifolds, such that, locally, this partition looks like the standard decomposition of ℝ^{n} into translates of ℝ^{d}. Ok, there's a caveat in my description since a same leaf could come infinitely many often in the neighborhood of a given point *m*. Anyways... I'm assuming that you know what a foliation is.

Foliations of *M* form a subset of distributions on *M*.

The **Frobenius integrability criterion** (mentioned by Tom in him remark) states that a distribution *D* comes from a foliation iff for any vector fields *v* and *w* tangent to *D*, their Lie bracket is again tangent to *D*.

It turns out that that criterion is *always* satisfied for one-dimensional distributions, and so one-dimensional distributions are indeed in bijection with one-dimensional foliations. But that's no longer true for *r* ≥ 2.

The operation of taking orthogonal complement is a very good operation for distributions: it's always well defined, and the orthogonal complement of the orthogonal complement is the distribution you started with.

But the orthogonal complement of a foliation is typically only a distribution.
The standard example that illustrates that situation is the vector field sin(*z*)*d/dx* + cos(*z*)*d/dy* on ℝ^{3}. It defines a perfectly good foliation, but its orthogonal fails to satisfy the Frobenius integrability criterion, and therefore fails to be a foliation (in this particular case, it's a contact structure, another beautiful mathematical notion...).

Ah! You also wanted the foliation to be defined by the orbits of a group acting by isometries... That can be arranged: take the action of *S*^{1} on *S*^{3} given by the Hopf fibration. The orthogonal distribution is the standard contact structure on *S*^{3}.

You also said that you wanted you Riemanninan manifold to be flat... In that case, you can take ℝ^{4}=ℂ^{2} with its *S*^{1}-action by complex multiplication. That example contains the above *S*^{3} example as an
invariant submanifold, and therefore reproduces all its features.