  We study differential systems for which it is possible to establish a
correspondence between symmetries and conservation laws based on Noether
identity: quasi-Noether systems. We analyze Noether identity and show that it
leads to the same conservation laws as Lagrange (Green-Lagrange) identity. We
discuss quasi-Noether systems, and some of their properties, and generate
classes of quasi-Noether differential equations of the second order. We next
introduce a more general version of quasi-Lagrangians which allows us to extend
Noether theorem. Here, variational symmetries are only sub-symmetries, not true
symmetries. We finally introduce the critical point condition for evolution
equations with a conserved integral, demonstrate examples of its compatibility,
and compare the invariant submanifolds of quasi-Lagrangian systems with those
of Hamiltonian systems.
