Symmetric Sets of Solutions to Differential Problems

The presence of a (Lie-point) symmetry for a differential equation leads naturally to the useful notions of symmetric sets of solutions, i.e. of sets which are mapped into themselves by the symmetry, and of orbits of solutions. We introduce the definition of partial symmetry, and show that the above notions may be preserved, although the symmetry is not exact. We consider the quite exceptional case of the Liouville equation, which admits an extremely large algebra of symmetries (the conformal symmetry algebra), and we shall see that any modification of this equation destroys this situation, but leaves the possibility of the existence of partial symmetries. Other simple examples are also considered, including a case of generalized (or Lie–Bäcklund) symmetry.