  We construct compact initial data of constant mean curvature $\widetilde{K}$
for Einstein's 4d vacuum equations with $\widehat{\Lambda} = \Lambda -
(\widetilde{K}^2/3)$ positive, where $\Lambda$ is the cosmological constant,
via the conformal method. To construct a transverse, trace-free (TT) momentum
tensor explicitly we first observe that, if the seed manifold has two
orthogonal Killing vectors, their symmetrized tensor product is a natural TT
candidate. Without the orthogonality requirement, but on locally conformally
flat seed manifolds there is a generalized construction for the momentum which
also involves the derivatives of the Killing fields found in work by Beig and
Krammer [2]. We consider in particular the round three sphere and classify the
TT tensors resulting from all possible pairs of its six Killing vectors,
focusing on the commuting case where the seed data are $\mathbb{U}(1) \times
\mathbb{U}(1)$ -symmetric. As to solving the Lichnerowicz equation, we discuss
in particular potential "symmetry breaking" by which we mean that solutions
have less symmetries than the equation itself; we compare with the case of the
"round donut" of topology $\mathbb{S}^2 \times \mathbb{S}$. In the absence of
symmetry breaking, the Lichnerowicz equation for a $\mathbb{U}(1) \times
\mathbb{U}(1)$ symmetric momentum on $\mathbb{S}^3$ reduces to an ODE. We
analyze distinguished families of solutions and the resulting data via a
combination of analytical and numerical techniques. Finally we investigate
marginally trapped surfaces of toroidal topology in our data.
