  In this work, we study a class of rotational surfaces in the pseudo-Euclidean
space $\mathbb{E}_2^4$ whose profile curves lie in two-dimensional planes. We
solve the differential equation that characterizes the rotational surfaces with
zero mean curvature to determine the profile curves of such rotational
surfaces. Then, we give some explicit parametrization of maximal rotational
surfaces and the timelike surfaces with zero mean curvature in
$\mathbb{E}_2^4$.
