Lie algebras of conservation laws of variational ordinary differential equations

We establish a new version of the first Noether Theorem, according to which the (equivalence classes of) first integrals of given Euler–Lagrange equations in one independent variable are in exact one-to-one correspondence with the (equivalence classes of) vector fields satisfying two simple geometric conditions, namely they simultaneously preserve the holonomy distribution of the jets space and the action from which the Euler–Lagrange equations are derived.