  We revisit the construction of maximal initial data on compact manifolds in
vacuum with positive cosmological constant via the conformal method. We
discuss, extend and apply recent results of Hebey et al. [19] and Premoselli
[31] which yield existence, non-existence, (non-)uniqueness and
(linearisation-) stability of solutions of the Lichnerowicz equation, depending
on its coefficients. We then focus on so-called $(t,\varphi)$-symmetric data as
"seed manifolds", and in particular on Bowen-York data on the round hypertorus
$\mathbb{S}^2 \times \mathbb{S}$ (a slice of Nariai) and on Kerr-deSitter. In
the former case, we clarify the bifurcation structure of the axially symmetric
solutions of the Lichnerowicz equation in terms of the angular momentum as
bifurcation parameter, using a combination of analytical and numerical
techniques. As to the latter example, we show how dynamical data can be
constructed in a natural way via conformal rescalings of Kerr-deSitter data.
