  Let M be a real analytic Riemannian manifold. An adapted complex structure on
TM is a complex structure on a neighborhood of the zero section such that the
leaves of the Riemann foliation are complex submanifolds. This structure is
called entire if it may be extended to the whole of TM. We call such manifolds
Grauert tubes, or simply tubes. We consider here the case of M = G a compact
connected Lie group with a left-invariant metric, and try to determine for
which such metrics the associated tube is entire. It is well-known that the
Grauert tube of a bi-invariant metric on a Lie group is entire. The case of the
smallest group SU(2) is treated completely, thanks to the complete
integrability of the geodesic flow for such a metric, a standard result in
classical mechanics. Along the way we find a new obstruction to tubes being
entire which is made visible by the complete integrability. (New reference and
exposition shortened, 11/17/2017.)
