= Solution
The infinitesimal connected <internal symmetry of a classical field theory> is $\delta\phi_1=\epsilon\phi_1$, $\delta\phi_2=-\epsilon\phi_2$. For this exact invariance of the <Lagrangian density>, the <Noether current> is $J^\mu=\sum_j\mathcal L_{\partial_\mu\phi_j}\delta\phi_j/\epsilon$. In coordinates $(t,x)$ it is
$$
\boxed{J^0=\phi_1\phi_{2,t}-\phi_2\phi_{1,t},\qquad J^x=c^2(\phi_2\phi_{1,x}-\phi_1\phi_{2,x}).}
$$
Varying the two fields gives the two <wave equations> $\phi_{1,tt}-c^2\phi_{1,xx}=0$ and $\phi_{2,tt}-c^2\phi_{2,xx}=0$. Direct differentiation cancels the first-derivative cross terms:
$$
\begin{aligned}
\partial_tJ^0+\partial_xJ^x
&=\phi_1(\phi_{2,tt}-c^2\phi_{2,xx})-\phi_2(\phi_{1,tt}-c^2\phi_{1,xx})\\
&=\boxed{0}.
\end{aligned}
$$
The <Noether charge> $Q=\int J^0dx$ is constant when the spatial boundary flux vanishes. The disconnected transformations from part (a) do not have independent infinitesimal generators of this type.
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