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