  Given a singular foliation, we attach an "essential isotropy" group to each
of its leaves, and show that its discreteness is the integrability obstruction
of a natural Lie algebroid over the leaf. We show that a condition ensuring
discreteness is the local surjectivity of a transversal exponential map
associated with the maximal ideal of vector fields prescribed to be tangent to
the foliation.
  The essential isotropy group is also shown to control the smoothness of the
holonomy cover of the leaf (the associated fiber of the holonomy groupoid), as
well as the {smoothness} of the associated isotropy group. Namely, the
(topological) closeness of the essential isotropy group is a necessary and
sufficient condition for the holonomy cover to be a smooth (finite-dimensional)
manifold and the isotropy group to be a Lie group.
  These results are useful towards understanding the normal form of a singular
foliation around a compact leaf. At the end of this article we briefly outline
work of ours on this normal form, to be presented in a subsequent paper.
