Future complete Einsteinian space times with U(1) symmetry, the unpolarized case

I prove the existence of vacuum $S^1$ symmetric Einsteinian, unpolarized, space times which are complete in the direction of the expansion, for small initial data, without supposing that the $S^1$ orbits are orthogonal to the 3-manifolds, as was done in previous work in collaboration with V. Moncrief.