  The canonical ADM equations are solved in terms of the conformal factor in
the instantaneous York gauge. A simple derivation is given for the solution of
the two body problem. A geometrical characterization is given for the apparent
singularities occurring in the N-body problem and it is shown how the Garnier
hamiltonian system arises in the ADM treatment by considering the time
development of the conformal factor at the locations where the extrinsic
curvature tensor vanishes. The equations of motion for the position of the
particles and of the apparent singularities and also the time dependence of the
linear residues at such singularities are given by the transformation induced
by an energy momentum tensor of a conformal Liouville theory. Such an equation
encodes completely the dynamics of the system.
