Finding the magnetic field by using Maxwell's laws (differential form)

Question

According to my physics book, one can use the following laws to identity the magnetic field:
div B = 0
rot B = μ0J
My book also states that, given a specific current density vector, many magnetic fields can satisfy the laws above, in corrispondente to the same current density vector. Esch field can be identified up to an uniform magnetic field. I didn't understand  properly this concept.

J. Murray · Accepted Answer

The author means that given a fixed current density $mathbf J(mathbf r)$ and a magnetic field $mathbf B(mathbf r)$ which satisfies
$$nabla cdot mathbf B = 0$$
$$nabla times mathbf B = mu_0 mathbf J$$
then if $mathbf B_0$ is a constant magnetic field (i.e. it is the same everywhere) then $mathbf B' = mathbf B + mathbf B_0$ also satisfies the two equations.  Therefore, you cannot start from those equations and uniquely determine the magnetic field.
In this way, it is very similar to the fact that an antiderivative is only determined up to some arbitrary integration constant.

Finding the magnetic field by using Maxwell's laws (differential form)

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