We present some methods of determining explicit solutions for self-dual supermembranes in 4 + 1 and 8 + 1 dimensions with spherical or toroidal topology. For configurations of axial symmetry, the continuous SU(infinity) Toda equation turns out to play a central role, and a specific method of determining all the periodic solutions are suggested. A number of examples are studied in detail. (C) 1998 Elsevier Science B.V.