Uniqueness of the definition of Noether current On page 28 of Pierre Ramond Field theory - A modern primer the following is written:

"we remark that a conserved current does not have a unique definition since we can always add to it the four-divergence of an antisymmetric tensor [...] Also since $j$ [the Noether current] is conserved only after use of the equations of motion we have the freedom to add to it any quantity which vanishes by virtue of the equations of motion".

I do not understand what he means by saying, any quantity which vanishes by virtue of the equations of motion.
 A: In Noether's first theorem, the continuity equation$^1$
$$ d_{\mu} J^{\mu}~\approx~0 \tag{*}$$
is an on-shell equation, i.e. it holds if the EOMs [= Euler-Lagrange (EL) equations] are satisfied. It does not necessarily hold off-shell.
Hence we can modify the Noether current $J^{\mu}$ with

*

*terms that vanish on-shell, and/or


*terms of the form $d_{\nu}A^{\nu\mu}$, where $A^{\nu\mu}=-A^{\mu\nu}$ is an antisymmetric tensor,
without spoiling the continuity eq. (*).
--
$^1$ The $\approx$ symbol means equality modulo EOMs.
A: I can give you an example:
$$
S=\int dt \left(\dot x^+\dot x^--x^+ x^-\right)
$$
has a conserved current associated with $x^{\pm}\rightarrow e^{\pm\rho} x^{\pm}$ given by
$$
j=x^+\dot x^--x^-\dot x^+
$$
This means that the current above will be conserved if the equations of motion are satisfied. Now if we add to this current a term of the form $\ddot x^++x^+$ the statement will stil be true, i.e. the current will stil be conserved. This is due to the fact that this term that I added is precisely the equations of motion.
