The geometric (physical) initial data is referred to as a triple (M,g,K) where (M,g) is a Riemannian 3-fold and K is a symmetric 2-tensor. They cannot be chosen freely; they must satisfy the constraints. Finding and studying solutions of the constraints are notoriously difficult. In this talk, we give a brief introduction to the standard conformal method, initiated by Lichnerowicz, and extended by Choquet-Bruhat and York. There is another way to construct vacuum initial data, referred to as 'the conformally covariant split' or, historically, 'Method B.' Amazingly, much less is mathematically known for this method. Joint with P.Mach and Y.Wang, we prove existence of solutions of the conformally covariant split system giving rise to non-constant mean curvature vacuum initial data for the Einstein field equations.

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