G. Cicogna , G. Gaeta
Partial Lie-point symmetries of differential equations
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ABSTRACT. When we consider a differential equation $\De=0$ 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 $\De=0$ is tranformed into a (generally, different) solution to
the same equation; here we define {\it partial} symmetries of $\De=0$ as
Lie-point invertible transformations $T$ such that there
is a nonempty subset $\cR \subset \S$ such that $T(\cR) = \cR$, i.e.
such that there is a subset of solutions to $\De=0$ which are transformed
one into the other. We discuss how to determine both partial symmetries
and the invariant set $\cR \subset \S$, 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; in the appendix we will discuss the case of
discrete partial symmetries.