  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.
