Authors’ summary: We prove existence of the solutions of the constraint equations satisfying “hyperboloidal boundary conditions” using the Choquet-Bruhat-Lichnerowicz-York conformal method and we analyze in detail their differentiability near the conformal boundary. We show that generic “hyperboloidal initial data” display asymptotic behaviour which is not compatible with Penrose’s hypothesis of smoothness of \(\mathcal S\). We also show that a large class of “non-generic” initial data satisfying Penrose smoothness conditions exists. The results are established by developing a theory of regularity up-to-boundary for a class of linear and semilinear elliptic systems of equations uniformly degenerating at the boundary.