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Using the implicit function theorem, we prove existence of solutions of theso-called conformally covariant split system on compact 3-dimensionalRiemannian manifolds. They give rise to non-Constant Mean Curvature (non-CMC)vacuum initial data for the Einstein equations. We investigate the conformallycovariant split system defined on compact manifolds with or without boundaries.In the former case, the boundary corresponds to an apparent horizon in theconstructed initial data. The case with a cosmological constant is thenconsidered separately. Finally, to demonstrate the applicability of theconformal covariant split system in numerical studies, we provide numericalexamples of solutions on manifolds $\mathbb S^1 \times \mathbb S^2$ and$\mathbb S^1 \times \mathbb T^2$.

Construction of vacuum initial data by the conformally covariant split system