= Solution
The <divergence of a cross product> identity and the vacuum <Maxwell equations> imply
$$
\begin{aligned}
\nabla\cdot(\mathbf B\times\mathbf E)
&=\mathbf E\cdot(\nabla\times\mathbf B)-\mathbf B\cdot(\nabla\times\mathbf E)\\
&=\frac1{c^2}\mathbf E\cdot\partial_t\mathbf E+\mathbf B\cdot\partial_t\mathbf B\\
&=\partial_t\left(\frac{|\mathbf E|^2}{2c^2}+\frac{|\mathbf B|^2}{2}\right).
\end{aligned}
$$
The <divergence theorem> therefore gives the required balance with
$$
\boxed{\lambda=\frac12\left(\frac{|\mathbf E|^2}{c^2}+|\mathbf B|^2\right)},\qquad
\boxed{\frac\partial{\partial t}\int_V\lambda\,dV=\oint_A(\mathbf B\times\mathbf E)\cdot d\mathbf A}.
$$
An additive time-independent constant in $\lambda$ is immaterial. This is the vacuum <Poynting theorem> in the normalization of the given equations: equivalently $\partial_t\lambda+\nabla\cdot(\mathbf E\times\mathbf B)=0$. The order of the <cross product> accounts for the plus sign in the stated integral identity.
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