  We consider the focusing energy critical NLS with inverse square potential in
dimension $d= 3, 4, 5$ with the details given in $d=3$ and remarks on results
in other dimensions. Solutions on the energy surface of the ground state are
characterized. We prove that solutions with kinetic energy less than that of
the ground state must scatter to zero or belong to the stable/unstable
manifolds of the ground state. In the latter case they converge to the ground
state exponentially in the energy space as $t\to \infty$ or $t\to -\infty$. (In
3-dim without radial assumption, this holds under the compactness assumption of
non-scattering solutions on the energy surface.) When the kinetic energy is
greater than that of the ground state, we show that all radial $H^1$ solutions
blow up in finite time, with the only two exceptions in the case of 5-dim which
belong to the stable/unstable manifold of the ground state. The proof relies on
the detailed spectral analysis, local invariant manifold theory, and a global
Virial analysis.
