A generalization of the Bowen–York initial data to the case with a positive cosmological constant is investigated. We follow the construction presented recently by Bizoń, Pletka and Simon, and solve numerically the Lichnerowicz equation on a compactified domain . In addition to two branches of solutions depending on the polar variable on that were already known, we find branches of solutions depending on two variables: the polar variable on and the coordinate on . Using Vanderbauwhede's results concerning bifurcations from symmetric solutions, we show the existence of the corresponding bifurcation points. By linearizing the Lichnerowicz equation and solving the resulting eigenvalue problem, we collect numerical evidence suggesting the absence of additional branches of solutions.