  This the first in a series of papers whose ultimate goal is to establish the
full nonlinear stability of the Kerr family for $|a|\ll m$. The paper builds on
the strategy laid out in \cite{KS} in the context of the nonlinear stability of
Schwarzschild for axially symmetric polarized perturbations. In fact the
central idea of \cite{KS} was the introduction and construction of generally
covariant modulated (GCM) hypersurfaces on which specific geometric quantities
take Schwarzschildian values. This was made possible by taking into account the
full general covariance of the Einstein vacuum equations. The goal of this
paper is to get rid of the symmetry restriction in the construction of GCM
spheres and thus remove an essential obstruction in extending the result of
\cite{KS} to a full stability proof of the Kerr family.
