Reductions: precontact versus presymplectic

We show that contact reductions can be described in terms of symplectic the traditional Marsden-Weinstein-Meyer as well constant rank picture. The point is we view structures particular (homogeneous) structures. A group action by contactomorphisms lifted to a Hamiltonian on corresponding manifold, called cover manifold. In contrast majority literature subject, our approach includes general (not only co-oriented) and changes point: Hamiltonians moment maps for contactomorphism groups are no longer defined manifold itself, but its cover. Actually, developed framework slightly more than purely contact, precontact presymplectic setting which based observation there one-to-one correspondence between isomorphism classes manifolds certain homogeneous manifolds.