Local and nonlocal solvable stuctures in the reduction of ODEs

Solvable structures, likewise solvable algebras of local
symmetries, can be used to integrate scalar ODEs by quadratures.
Solvable structures, however, are particularly suitable for the
integration of ODEs with a lack of local symmetries. In fact,
under regularity assumptions, any given ODE always admits solvable
structures even though finding them in general could be a very
difficult task. In practice a noteworthy simplification may come
by computing solvable structures which are adapted to some
admitted symmetry algebra. In this paper we consider solvable
structures adapted to local and nonlocal symmetry algebras of any
order (i.e., classical and higher). In particular we introduce the
notion of nonlocal solvable structure.