We use group theoretic methods to obtain the extended Lie point symmetries of
the equations of motion for a charged particle in the field of a monopole.
Cases with certain model magnetic fields and potentials are also studied. Our
analysis gives the generators and Lie algebras generating the inherent
symmetries. The equations of motion of a scalar particle probing the near
horizon structure of a black hole is also treated likewise. We have also found
the generators of Krause's complete symmetry groups for some of the above
examples.