Maximal surfaces, a fascinating class of surfaces in differential geometry, are identified by having a mean curvature equal to zero. This distinctive feature gives rise to a nonlinear second-order partial differential equation. In this current article, we delve into the symmetries that underlie the maximal surface equation. Next, we identify a one-dimensional optimal system of subalgebras that span these symmetries. It provides a powerful tool to analyze and manipulate the equation, making it easier to study. Finally, since we aim not only to explore the underlying symmetries of the maximal surface equation, we demonstrate how these symmetries can be harnessed to uncover and classify a wide range of maximal surfaces by using reduction methods.
