Non-Noether symmetries in singular dynamical systems

It's well known that Noether symmetries lead to the conservation laws. Conserved quantities are constructed out of generator of the symmetry - invariant Hamiltonian vector field. Considering more general class of vector fields - non-Hamiltonian ones leads to the notion of non-Noether symmetry and conservation laws (Lutzky's integrals of motion) with interesting properties. In the present paper correspondence between non-Noether symmetries and conserved quantities in different types of dynamical systems (DS on symplectic, presymplectic and Poisson manifolds) is considered.