Summary. We give a geometrical interpretation of the notion of µ-prolongations of vector fields and of the related concept of µ-symmetry for partial differential equations (extending to PDEs the notion of λ-symmetry for ODEs). We give in particular a result concerning the relationship between µ-symmetries and standard exact symmetries. The notion is also extended to the case of conditional and partial symmetries, and we analyze the relation between local µ-symmetries and nonlocal standard symmetries.

When we consider a differential equation ∆ = 0 whose set of solutions is S∆, a Lie-point exact symmetry of this is a Lie-point invertible transformation T such that T (S∆) = S∆, i.e. such that any solution to ∆ = 0 is tranformed into a (generally, different) solution to the same equation; here we define partial symmetries of ∆ = 0 as Lie-point invertible transformations T such that there is a nonempty subset P ⊂ S∆ such that T (P) = P, i.e. such that there is a subset of solutions to ∆ = 0 which are transformed one into the other. We discuss how to determine both partial symmetries and the invariant set P ⊂ S∆, 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 indeed extended to partial generalized (or Lie-Bäcklund) symmetries along the same lines, and in the appendix we will discuss the case of discrete partial symmetries.

Abstract. We consider a deformation of the prolongation operation, defined on sets of vector fields and involving a mutual interaction in the definition of prolonged ones. This maintains the "invariants by differentiation" property, and can hence be used to reduce ODEs satisfying suitable invariance conditions in a fully algorithmic way, similarly to what happens for standard prolongations and symmetries.PACS numbers: 02.30Hq. MSC numbers: 34C14; 34A34, 34A05, 53A55

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We give a version of Noether theorem adapted to the framework of µ-symmetries; this extends to such case recent work by Muriel, Romero and Olver in the framework of λ-symmetries, and connects µ-symmetries of a Lagrangian to a suitably modified conservation law. In some cases this "µconservation law" actually reduces to a standard one; we also note a relation between µ-symmetries and conditional invariants. We also consider the case where the variational principle is itself formulated as requiring vanishing variation under µ-prolonged variation fields, leading to modified Euler-Lagrange equations. In this setting µ-symmetries of the Lagrangian correspond to standard conservation laws as in the standard Noether theorem. We finally propose some applications and examples. Reduction of µ-prolongation to ordinary and λ onesWe can write the coefficients ψ a J of the µ-prolongation (8) aswhere ϕ a J are the coefficients of the standard prolongation (2,3); inspection of the standard and µ-prolongation formulas in recursive form (see [14] for details) shows that the recursion formula for the µ-difference terms F a J is

The notion of λ-symmetries, originally introduced by C. Muriel and J.L. Romero, is extended to the case of systems of first-order ODE's (and of dynamical systems in particular). It is shown that the existence of a symmetry of this type produces a reduction of the differential equations, restricting the presence of the variables involved in the problem. The results are compared with the case of standard (i.e. exact) Lie-point symmetries and are also illustrated by some examples.

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