The normal tagline for energy conservation is that it's a conserved quantity associated to time-translation invariance. I understand how this works for theories coming from a Lagrangian, and that this is the context that the above statement is intended to refer to, but I'm curious as to whether or not it's true in greater generality (i.e. is true in a wider context than can be shown through Noether's theorem). I'll stick to single ODEs, since this case is already unclear to me. If we have a differential equation
$$\ddot{x}=f(x,\dot{x})$$
for general $f$ this clearly possesses time translation symmetry. This does not conserve energy as the term is normally used, since this includes examples such as a damped harmonic oscillator. However is there actually no conserved quantity of any kind associated to the symmetry? If there's no dependence on $\dot{x}$ we can easily find an integral of motion, but I'm not sure why any dependence on $\dot{x}$ would ruin this.
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