A contribution to the symmetry classification problem for second-order PDEs in z(x, y)

I report on a contribution to the point symmetry classification problem for second-order partial differential equations (PDEs) in z(x, y), i.e. to an overview over all possible symmetry groups admitted by this class of equations. The article also contains a concise introduction into classical symmetry analysis. Sophus Lie (1842–1899) determined all continuous transformation groups of the 2D plane and gave normal forms for any ordinary differential equation that is invariant under one of those groups. I deal with the extension of Lie's program to second-order PDEs in z(x, y). The starting point to this endeavour is a previously unknown paper by Amaldi from 1901, which claims to have completed Lie's classification of groups acting in (x, y, z)-space. I also present a Maple procedure (‘LHSO1_PDE_Solver’) for solving systems of linear, homogeneous first-order PDEs that performs better on this class than Maple's built-in PDE system solver.