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We construct solutions to the constraint equations in general relativityusing the limit equation criterion introduced by Dahl, Humbert and the firstauthor. We focus on solutions over compact 3-manifolds admitting a$\bS^1$-symmetry group. When the quotient manifold has genus greater than 2, weobtain strong far from CMC results.

Limit equation for vacuum Einstein constraints with a translational Killing vector field in the compact hyperbolic case