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 (2008 Commun. Math. Phys. 278 117) and Premoselli (2015 Calc. Var. 53 29–64) which yield existence, non-existence, (non-)uniqueness and (linearization-) stability of solutions of the Lichnerowicz equation, depending on its coefficients. We then focus on so-called ()-symmetric data as 'seed manifolds', and in particular on Bowen–York data on the round hypertorus (a slice of Nariai) and on Kerr–deSitter (KdS). 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 a 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 KdS data.