in the non-linear regime. Bowen and York [@BoYo] (and with a different set of boundary conditions more recently Brandt and Brügmann [@BrBr]) studied the problem of giving initial data for boosted and spinning holes in such a way that a superposition is possible. They assume the spatial metric is conformally flat and as a consequence one can superpose the extrinsic curvatures for the two holes and still solve the momentum constraint. One then proceeds to find a conformally flat spatial metric for the superposed holes by solving a nonlinear elliptic partial differential equation. The procedure achieves superposition at the price of assuming conformal flatness of the three metrics, which is not generically possible, and more importantly, is not possible in situations of interest. For instance a single spinning black hole, described by the Kerr solution, is not known to admit conformally flat spatial slices [@BoYo; @GaPr].

A similar situation develops for the case of a single boosted black hole. The initial data constructed by Bowen and York or Brandt and Brügmann do not correspond to those one would find on a boosted slice of the Schwarzschild spacetime. In this paper we will refer to these families as “conformal boosted black 