  In this paper we provide a family of algebraic space-like surfaces in the
three dimensional anti de Sitter space that shows that this Lorentzian manifold
admits algebraic maximal examples of any order. Then, we classify all the
space-like order two algebraic maximal hypersurfaces in the anti de Sitter
$N$-dimensional space. Finally, we provide two families of examples of
Lorentzian order two algebraic zero mean curvature in the de Sitter space.
