  We discuss geometric properties of non-Noether symmetries and their possible
applications in integrable Hamiltonian systems. Correspondence between
non-Noether symmetries and conservation laws is revisited. It is shown that in
regular Hamiltonian systems such a symmetries canonically lead to a Lax pairs
on the algebra of linear operators on cotangent bundle over the phase space.
Relationship 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. Action of one-parameter group of symmetry on
algebra of integrals of motion is studied and involutivity of group orbits is
discussed. Hidden non-Noether symmetries of Toda chain, nonlinear Schrodinger
equation, Korteweg-de Vries equations, Benney system, nonlinear water wave
equations and Broer-Kaup system are revealed and discussed.
