  Let $D$ be a bounded logarithmically convex complete Reinhardt domain in
$\mathbb{C}^n$ centered at the origin. Generalizing a result for the
one-dimensional case of the unit disk, we prove that the $C^*$-algebra
generated by Toeplitz operators with bounded measurable separately radial
symbols (i.e., symbols depending only on $|z_1|$, $|z_2|$, ..., $|z_n|$) is
commutative.
  We show that the natural action of the $n$-dimensional torus $\mathbb{T}^n$
defines (on a certain open full measure subset of $D$) a foliation which
carries a transverse Riemannian structure having distinguished geometric
features. Its leaves are equidistant with respect to the Bergman metric, and
the orthogonal complement to the tangent bundle of such leaves is integrable to
a totally geodesic foliation. Furthermore, these two foliations are proved to
be Lagrangian.
  We specify then the obtained results for the unit ball.
