= Solution
For translation in a fixed direction $\hat{\mathbf n}$, take $\mathbf K=\hat{\mathbf n}$ and the preceding $F_{\mathbf n}$. The triple-product identity gives $F_{\mathbf n}=-(q/2)\hat{\mathbf n}\cdot(\mathbf B\times\mathbf x)$. Thus
$$
Q_{\mathbf n}=\mathbf p\cdot\hat{\mathbf n}-F_{\mathbf n}
=\hat{\mathbf n}\cdot\left(m\dot{\mathbf x}+q\mathbf B\times\mathbf x\right).
$$
Choosing the three Cartesian directions gives the three constants
$$
\boxed{\mathbf K_{\rm mag}=m\dot{\mathbf x}+q\mathbf B\times\mathbf x
=\text{constant vector}.}
$$
These are magnetic pseudomomentum components, not the three components of kinetic momentum $m\dot{\mathbf x}$. A direct check is $d\mathbf K_{\rm mag}/dt=q\dot{\mathbf x}\times\mathbf B+q\mathbf B\times\dot{\mathbf x}=0$. In the stated gauge, $\mathbf K_{\rm mag}=\mathbf p+(q/2)\mathbf B\times\mathbf x$, consistent with the Noether boundary correction.
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