Take the Lagrangian $L=\frac{1}{2}m{{\left( \frac{{\rm{d}}}{{\rm{d}}t}x \right)}^{2}}-\frac{1}{2}k{{x}^{2}}$, for example.
The equation of motion of this system should be given by $m\frac{{{{\rm{d}}}^{2}}}{{\rm{d}}{{t}^{2}}}x+kx=0$.
Now considering the transform $x\to x+\varepsilon$.
The Lagrangian is not invariant under this transform so there is no reservation of momentum, but I'm not so sure whether the equation of motion is "covariant" under this transform.
