So, if we have system of differential equations obtained from Lagrange function, by means of Noether theoerem (if we know some one-parameter symmetry group), we can derive conserved quantity.


But how does it work with arbitrary system (not derived from Euler-Lagrange equations).
Does it mean that knowledge of symmetry give us somehow conserved quantity?



Conserved quantities
If the dynamics does not have a variational formulation a conservation law is not necessarily related to a symmetry. A more general approach than starting from the Euler-Lagrange equations is described in [Construction of conservation laws: how the direct method generalizes Noether’s theorem](https://personal.math.ubc.ca/~bluman/cyprus%20proceedings%20paper.pdf).


