  In this paper we devise a systematic procedure to obtain nonlocal symmetries
of a class of scalar nonlinear ordinary differential equations (ODEs) of
arbitrary order related to linear ODEs through nonlocal relations. The
procedure makes use of the Lie point symmetries of the linear ODEs and the
nonlocal connection to deduce the nonlocal symmetries of the corresponding
nonlinear ODEs. Using these nonlocal symmetries we obtain reduction
transformations and reduced equations to specific examples. We find the reduced
equations can be explicitly integrated to deduce the general solutions for
these cases. We also extend this procedure to coupled higher order nonlinear
ODEs with specific reference to second order nonlinear ODEs.
