  We prove that the outer Lipschitz geometry of a germ $(X,0)$ of a normal
complex surface singularity determines a large amount of its analytic
structure. In particular, it follows that any analytic family of normal surface
singularities with constant Lipschitz geometry is Zariski equisingular. We also
prove a strong converse for families of normal complex hypersurface
singularities in $\mathbb C^3$: Zariski equisingularity implies Lipschitz
triviality. So for such a family Lipschitz triviality, constant Lipschitz
geometry and Zariski equisingularity are equivalent to each other.
