An integral constraint vector or ICV on a spacelike hypersurface \(\Sigma\) in a space-time is a vector field tangent to \(\Sigma\) satisfying two differential equations involving the first and the second fundamental forms of \(\Sigma\). The author interprets an ICV as a covariant constant object with respect to a connection depending on \(\Sigma\). So, the existence of ten linearly independent ICVs is reduced to the finding of the conditions for the vanishing of the curvature of this connection. These conditions are interpreted in terms of embeddability of \(\Sigma\) in a space of constant curvature. Finally, the ICVs are related to three- surface twistors.