  We test the robustness of the conditions required for the existence of
(supersymmetric) warped flux anti-de Sitter, de Sitter, and Minkowski
backgrounds in supergravity theories using as examples suitable foliations of
anti-de Sitter spaces. We find that there are supersymmetric de Sitter
solutions in supergravity theories including maximally supersymmetric ones in
10- and 11-dimensional supergravities. Moreover, warped flux Minkowski
backgrounds can admit Killing spinors which are not Killing on the Minkowski
subspace and therefore cannot be put in a factorized form.
