  We disscuss some geometric aspects of the concept of non-Noether symmetry. It
is shown that in regular Hamiltonian systems such a symmetry canonically leads
to a Lax pair on the algebra of linear operators on cotangent bundle over the
phase space. Correspondence between the non-Noether symmetries and other wide
spread geometric methods of generating conservation laws such as bi-Hamiltonian
formalism, bidifferential calculi and Frolicher-Nijenhuis geometry is
considered. It is proved that the integrals of motion associated with the
continuous non-Noether symmetry are in involution whenever the generator of the
symmetry satisfies a certain Yang-Baxter type equation.
