Partial Lie-point symmetries of differential equations

When we consider a differential equation Delta = 0 whose set of solutions i s S-Delta, a Lie-point exact symmetry of this is a Lie-point invertible tra nsformation T such that T(S-Delta) = S-Delta, i.e. such that any solution t o Delta = 0 is transformed into a (generally, different) solution to the sa me equation; here we define partial symmetries of Delta = 0 as Lie-point in vertible transformations T such that there is a non-empty subset P subset o f S-Delta such that T (P) = P, i.e. such that there is a subset of solution s to Delta = 0 which are transformed into one another. We discuss how to de termine both partial symmetries and the invariant set P subset of S-Delta, and show that our procedure is effective by means of concrete examples. We also discuss relations with conditional symmetries, and how our discussion applies to the special case of dynamical systems. Our discussion will focus on continuous Lie-point partial symmetries, but our approach would also be suitable for more general classes of transformations; the discussion is in deed extended to partial generalized (or Lie-Backlund) symmetries along the same lines, and in the appendix we will discuss the case of discrete parti al symmetries.