Symmetries and differential equations

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Geometrical symmetries correspond to transformations of geometrical objects that leave them invariant.  For example, a square is invariant under discrete rotations by 90 degrees about its centre. A circle is invariant under a continuous symmetry corresponding to rotations by any angle about its centre.  The invariance of a differential equation under a continuous symmetry can lead to its solutions, its transformation to a simpler equation and/or to conservation laws.  Often one can determine obvious symmetries such as invariance under rotations or scalings due to dimensional reasoning.  In recent years the use of symbolic computation with high speed computers allows one to effectively utilize and extend algorithms to find and apply continuous symmetries not obvious by inspectional analysis.  This proposal is concerned with finding further extensions.