An extension of the Noether theorem: Accompanying equations possessing conservation laws

•Noether’s theorem on conservation laws is extended by adding accompanying equations.•It is shown that each Noether symmetry gives a conservation law for accompanying equations.•The method is applied to the 2nd-order ODEs from Lie’s group classification.•Similar construction is made for canonical Hamiltonian equations. It is shown that the Noether theorem can be extended for some equations associated (accompanying) with Euler–Lagrange equation. Each symmetry of Lagrangian yields a class of accompanying equations possessing conservation law (first integral). The generalization is done for canonical Hamiltonian equations as well.