Hidden symmetries and nonlocal group generators for ordinary differential

Hidden symmetries of ordinary differential equations (ODEs) are studied with nonlocal group generators. General forms are given for an exponential nonlocal group generator of an ODE that is reduced from a higher-order ODE, which is expressed in canonical variables and which is invariant under a two-parameter Lie group. The nonlocal group generator identifies a type I hidden symmetry. Type II hidden symmetries are found in some reduction pathways of an ODE invariant under a solvable, nonabelian, three-parameter Lie group. The algorithm for the appearance of the type II hidden symmetry is stated. General forms for the reduced nonlocal group generator, which identifies the type II hidden symmetry, are presented when the other two commuting original group generators are in normal form.