= Solution
Let $\widehat b=B/|B|$. For a constant uniform <magnetic field>, choose the symmetric gauge
$$
A=\frac12B\times r,
$$
which has $\nabla\times A=B$ and $A\cdot\widehat b=0$. With $E=0$, one may take $\phi=0$. The coordinate $r_\parallel=r\cdot\widehat b$ is then absent from the Lagrangian, so it is an <ignorable coordinate>. Its <generalized momentum>
$$
p_\parallel
=\frac{\partial L}{\partial\dot r_\parallel}
=(m\dot r+qA)\cdot\widehat b
=m\dot r\cdot\widehat b
$$
is conserved. Hence
$$
\boxed{\frac d{dt}(p\cdot\widehat b)=0},
$$
which is the <canonical momentum parallel to a uniform magnetic field> in this gauge.
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